https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Omer&feedformat=atomUW-Math Wiki - User contributions [en]2020-09-19T23:11:18ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19587Graduate Logic Seminar2020-08-27T15:32:58Z<p>Omer: Removed old schedules</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarily original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' on line (ask for code).<br />
* '''Organizers:''' [https://www.math.wisc.edu/~jgoh/ Jun Le Goh]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Fall 2020 - Tentative schedule ==<br />
<br />
=== September 14 - Josiah Jacobsen-Grocott ===<br />
<br />
Title, abstract TBA<br />
<br />
=== September 21 - Alice Vidrine ===<br />
<br />
Title, abstract TBA<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar,_previous_semesters&diff=19586Graduate Logic Seminar, previous semesters2020-08-27T15:32:15Z<p>Omer: Added Fall 19 and Spring 20 seminars</p>
<hr />
<div>This is an historic listing of the talks in the [[Graduate Logic Seminar]].<br />
<br />
== Spring 2020 ==<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
Title: Algebraic logic and algebraizable logics<br />
<br />
Abstract: Arguably the oldest theme in what we would recognize as "mathematical logic" is the algebraic interpretation of logic, the most famous example of this being the connection between (classical) propositional logic and Boolean algebras. But underlying the subject of algebraic logic is the implicit assumption that many logical systems are "satisfyingly" interpreted as algebraic structures. This naturally hints at a question, which to my knowledge went unasked for a surprisingly long time: when does a logic admit a "nice algebraic interpretation?"<br />
<br />
Perhaps surprisingly, this is actually a question which can be made precise enough to treat with interesting results. I'll sketch what is probably the first serious result along these lines, due to Blok and Pigozzi, and then say a bit about where this aspect of algebraic logic has gone from there.<br />
<br />
=== '''(Covid-19) Due to the cancellation of face-to-face instruction in UW-Madison through at least April 10, the seminar is suspended until further notice''' ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks - Iván Ongay Valverde and James Earnest Hanson ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks - Manlio Valenti and Patrick Nicodemus ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
== Spring 2018 ==<br />
<br />
=== January 29, Organizational meeting ===<br />
<br />
This day we decided the schedule for the semester.<br />
<br />
=== February 5, [http://www.math.wisc.edu/~andrews/ Uri Andrews] ===<br />
<br />
Title: Building Models of Strongly Minimal Theories - Part 1<br />
<br />
Abstract: Since I'm talking in the Tuesday seminar as well, I'll use the Monday seminar talk to do some background on the topic and some<br />
lemmas that will go into the proofs in Tuesday's talk. There will be (I hope) some theorems of interest to see on both days, and both on<br />
the general topic of answering the following question: What do you need to know about a strongly minimal theory in order to compute<br />
copies of all of its countable models. I'll start with a definition for strongly minimal theories and build up from there.<br />
<br />
=== February 12, James Hanson ===<br />
<br />
Title: Finding Definable Sets in Continuous Logic<br />
<br />
Abstract: In order to be useful the notion of a 'definable set' in continuous logic is stricter than a naive comparison to discrete logic<br />
would suggest. As a consequence, even in relatively tame theories there can be very few definable sets. For example, there is a<br />
superstable theory with no non-trivial definable sets. As we'll see, however, there are many definable sets in omega-stable,<br />
omega-categorical, and other small theories.<br />
<br />
=== February 19, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] ===<br />
<br />
Title: Proper forcing<br />
<br />
Abstract: Although a given forcing notion may have nice properties on its own, those properties might vanish when we apply it repeatedly.<br />
Early preservation results (that is, theorems saying that the iteration of forcings with a nice property retains that nice property)<br />
were fairly limited, and things really got off the ground with Shelah's invention of "proper forcing." Roughly speaking, a forcing is<br />
proper if it can be approximated by elementary submodels of the universe in a particularly nice way. I'll define proper forcing and<br />
sketch some applications. <br />
<br />
=== February 26, Patrick Nicodemus ===<br />
<br />
Title: A survey of computable and constructive mathematics in economic history<br />
<br />
=== March 5, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] ===<br />
<br />
Title: Convexly Orderable Groups <br />
<br />
=== March 12, [https://math.nd.edu/people/visiting-faculty/daniel-turetsky/ Dan Turetsky] (University of Notre Dame) ===<br />
<br />
Title: Structural Jump<br />
<br />
=== March 19, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] ===<br />
<br />
Title: Networks and degrees of points in non-second countable spaces<br />
<br />
=== April 2, Wil Cocke ===<br />
<br />
Title: Characterizing Finite Nilpotent Groups via Word Maps<br />
<br />
Abstract: In this talk, we will examine a novel characterization of finite nilpotent groups using the probability distributions induced by word maps. In particular we show that a finite group is nilpotent if and only if every surjective word map has fibers of uniform size.<br />
<br />
=== April 9, Tejas Bhojraj ===<br />
<br />
Title: Quantum Randomness<br />
<br />
Abstract: I will read the paper by Nies and Scholz where they define a notion of algorithmic randomness for infinite sequences of quantum bits (qubits). This talk will cover the basic notions of quantum randomness on which my talk on Tuesday will be based. <br />
<br />
=== April 16, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde] ===<br />
<br />
Title: What can we say about sets made by the union of Turing equivalence classes?<br />
<br />
Abstract: It is well known that given a real number x (in the real line) the set of all reals that have the same Turing degree (we will call this a Turing equivalence class) have order type 'the rationals' and that, unless x is computable, the set is not a subfield of the reals. Nevertheless, what can we say about the order type or the algebraic structure of a set made by the uncountable union of Turing equivalence classes?<br />
<br />
This topic hasn't been deeply studied. In this talk I will focus principally on famous order types and answer whether they can be achieved or not. Furthermore, I will explain some possible connections with the automorphism problem of the Turing degrees.<br />
<br />
This is a work in progress, so this talk will have multiple open questions and opportunities for feedback and public participation.(hopefully).<br />
<br />
=== April 23, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] (Thesis Defense) Start 3:45 Room B231===<br />
<br />
Title: Cototal enumeration degrees and their applications to effective mathematics <br />
<br />
Abstract: The enumeration degrees measure the relative computational difficulty of enumerating sets of natural numbers. Unlike the Turing degrees, the enumeration degrees of a set and its complement need not be comparable. A set is total if it is enumeration above its complement. Taken together, the enumeration degrees of total sets form an embedded copy of the Turing degrees within the enumeration degrees. A set of natural numbers is cototal if it is enumeration reducible to its complement. Surprisingly, the degrees of cototal sets, the cototal degrees, form an intermediate structure strictly between the total degrees and the enumeration degrees. <br />
<br />
Jeandel observed that cototal sets appear in a wide class of structures: as the word problems of simple groups, as the languages of minimal subshifts, and more generally as the maximal points of any c.e. quasivariety. In the case of minimal subshifts, the enumeration degree of the subshift's language determines the subshift's Turing degree spectrum: the collection of Turing degrees obtained by the points of the subshift. We prove that cototality precisely characterizes the Turing degree spectra of minimal subshifts: the degree spectra of nontrivial minimal subshifts are precisely the cototal enumeration cones. On the way to this result, we will give several other characterizations of the cototal degrees, including as the degrees of maximal anti-chain complements on <math>\omega^{<\omega}</math>, and as the degrees of enumeration-pointed trees on <math>2^{<\omega}</math>, and we will remark on some additional applications of these characterizations.<br />
<br />
=== April 30, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde]===<br />
<br />
Title: Definibility of the Frobenius orbits and an application to sets of rational distances.<br />
<br />
Abstract: In this talk I'll present a paper by Hector Pastén. We will talk about how having a formula that identify a Frobenius orbits can help you show an analogue case of Hilbert's tenth problem (the one asking for an algorithm that tells you if a diophantine equation is solvable or not).<br />
<br />
Finally, if time permits, we will do an application that solves the existence of a dense set in the plane with rational distances, assuming some form of the ABC conjecture. This last question was propose by Erdös and Ulam.<br />
<br />
== Fall 2017 ==<br />
<br />
=== September 11, Organizational meeting ===<br />
<br />
This day we decided the schedule for the semester.<br />
<br />
=== September 18, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] ===<br />
<br />
Title: The Kunen inconsistency<br />
<br />
Abstract: While early large cardinal axioms were usually defined combinatorially - e.g., cardinals satisfying a version of Ramsey's<br />
theorem - later focus shifted to model-theoretic definitions, specifically definitions in terms of elementary embeddings of the<br />
whole universe of sets. At the lowest level, a measurable cardinal is one which is the least cardinal moved (= critical point) by a<br />
nontrivial elementary embedding from V into some inner model M.<br />
<br />
There are several variations on this theme yielding stronger and stronger large cardinal notions; one of the most important is the<br />
inclusion of *correctness properties* of the target model M. The strongest such correctness property is total correctness: M=V. The<br />
critical point of an elementary embedding from V to V is called a *Reinhardt cardinal*. Shortly after their introduction in Reinhardt's<br />
thesis, however, the existence of a Reinhardt cardinal was shown to be inconsistent with ZFC.<br />
<br />
I'll present this argument, and talk a bit about the role of choice. <br />
<br />
=== September 25, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] ===<br />
<br />
Title: Hindman's theorem via ultrafilters<br />
<br />
Abstract: Hindman's theorem is a Ramsey-type theorem in additive combinatorics: if we color the natural numbers with two colors, there is an infinite set such that any *finite sum* from that set has the same color as any other finite sum. There are (to my knowledge) two proofs of Hindman's theorem: one of them is a complicated mess of combinatorics, and the other consists of cheating wildly. We'll do.<br />
<br />
=== October 2, James Hanson ===<br />
<br />
Title: The Gromov-Hausdorff metric on type space in continuous logic<br />
<br />
Abstract: The Gromov-Hausdorff metric is a notion of the 'distance' between two metric spaces. Although it is typically studied in the context of compact or locally compact metric spaces, the definition is sensible even when applied to non-compact metric spaces, but in that context it is only a pseudo-metric: there are non-isomorphic metric spaces with Gromov-Hausdorff distance 0. This gives rise to an equivalence relation that is slightly coarser than isomorphism. There are continuous first-order theories which are categorical with regards to this equivalence relation while failing to be isometrically categorical, so it is natural to look for analogs of the Ryll-Nardzewski theorem and Morley's theorem, but before we can do any of that, it'll be necessary to learn about the "topometric" structure induced on type space by the Gromov-Hausdorff metric.<br />
<br />
=== October 9, James Hanson ===<br />
<br />
Title: Morley rank and stability in continuous logic<br />
<br />
Abstract: There are various ways of counting the 'size' of subsets of metric spaces. Using these we can do a kind of Cantor-Bendixson analysis on type spaces in continuous first-order theories, and thereby define a notion of Morley rank. More directly we can define<br />
> the 'correct' notion of stability in the continuous setting. There are also natural Gromov-Hausdorff (GH) analogs of these notions. With this we'll prove that inseparably categorical theories have atomic models over arbitrary sets, which is an important step in the proof of Morley's theorem in this setting. The same proof with essentially cosmetic changes gives that inseparably GH-categorical theories have 'GH-atomic' models over arbitrary sets, but GH-atomic models fail to be GH-unique in general.<br />
<br />
=== October 23, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] ===<br />
<br />
Title: Boxy sets in ordered convexly-orderable structures.<br />
<br />
=== October 30, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde] ===<br />
<br />
Title: Dancing SCCA and other Coloring Axioms<br />
<br />
Abstract: In this talk I will talk about some axioms that are closely related to SOCA (Semi Open Coloring Axiom), being the main protagonist SCCA (Semi Clopen Coloring Axiom). I will give a motivation on the statements of both axioms, a little historic perspective and showing that both axioms coincide for separable Baire spaces. This is a work in progress, so I will share some open questions that I'm happy to discuss.<br />
<br />
=== November 6, Wil Cocke ===<br />
<br />
Title: Two new characterizations of nilpotent groups<br />
<br />
Abstract: We will give two new characterizations of finite nilpotent groups. One using information about the order of products of elements of prime order and the other using the induced probability distribution from word maps.<br />
<br />
Or...<br />
<br />
Title: Centralizing Propagating Properties of Groups<br />
<br />
Abstract: We will examine some sentences known to have finite spectrum when conjoined with the theory of groups. Hopefully we will be able to find new examples. <br />
<br />
=== November 13, [https://www.math.wisc.edu/~lempp/ Steffen Lempp] ===<br />
<br />
Title: The computational complexity of properties of finitely presented groups<br />
<br />
Abstract: I will survey index set complexity results on finitely presented groups.<br />
<br />
=== November 20, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] ===<br />
<br />
Title: Strong Difference Randomness<br />
<br />
Abstract: The difference randoms were introduced by Franklin and Ng to characterize the incomplete Martin-Löf randoms. More recently, Bienvenu and Porter introduced the strong difference randoms, obtained by imposing the Solovay condition over the class of difference tests. I will give a Demuth test characterization of the strong difference randoms, along with a lowness characterization of them among the Martin-Löf randoms. <br />
<br />
=== December 4, Tejas Bhojraj ===<br />
<br />
Title: Quantum Algorithmic Randomness<br />
<br />
Abstract: I will discuss the recent paper by Nies and Scholz where they define quantum Martin-Lof randomness (q-MLR) for infinite sequences of qubits. If time permits, I will introduce the notion of quantum Solovay randomness and show that it is equivalent to q-MLR in some special cases.<br />
<br />
=== December 11, Grigory Terlov ===<br />
<br />
Title: The Logic of Erdős–Rényi Graphs</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19251Graduate Logic Seminar2020-03-12T17:41:57Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
Title: Algebraic logic and algebraizable logics<br />
<br />
Abstract: Arguably the oldest theme in what we would recognize as "mathematical logic" is the algebraic interpretation of logic, the most famous example of this being the connection between (classical) propositional logic and Boolean algebras. But underlying the subject of algebraic logic is the implicit assumption that many logical systems are "satisfyingly" interpreted as algebraic structures. This naturally hints at a question, which to my knowledge went unasked for a surprisingly long time: when does a logic admit a "nice algebraic interpretation?"<br />
<br />
Perhaps surprisingly, this is actually a question which can be made precise enough to treat with interesting results. I'll sketch what is probably the first serious result along these lines, due to Blok and Pigozzi, and then say a bit about where this aspect of algebraic logic has gone from there.<br />
<br />
=== March 16 - Spring break - No seminar ===<br />
<br />
=== '''Due to the cancellation of face-to-face instruction in UW-Madison through at least April 10, the seminar is suspended until further notice''' ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19250Graduate Logic Seminar2020-03-12T17:41:19Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
Title: Algebraic logic and algebraizable logics<br />
<br />
Abstract: Arguably the oldest theme in what we would recognize as "mathematical logic" is the algebraic interpretation of logic, the most famous example of this being the connection between (classical) propositional logic and Boolean algebras. But underlying the subject of algebraic logic is the implicit assumption that many logical systems are "satisfyingly" interpreted as algebraic structures. This naturally hints at a question, which to my knowledge went unasked for a surprisingly long time: when does a logic admit a "nice algebraic interpretation?"<br />
<br />
Perhaps surprisingly, this is actually a question which can be made precise enough to treat with interesting results. I'll sketch what is probably the first serious result along these lines, due to Blok and Pigozzi, and then say a bit about where this aspect of algebraic logic has gone from there.<br />
<br />
=== March 16 - Spring break - No seminar ===<br />
<br />
'''<br />
== Due to the cancellation of face-to-face instruction in UW-Madison through at least April 10, the seminar is suspended until further notice ==<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19247Graduate Logic Seminar2020-03-12T17:40:46Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
Title: Algebraic logic and algebraizable logics<br />
<br />
Abstract: Arguably the oldest theme in what we would recognize as "mathematical logic" is the algebraic interpretation of logic, the most famous example of this being the connection between (classical) propositional logic and Boolean algebras. But underlying the subject of algebraic logic is the implicit assumption that many logical systems are "satisfyingly" interpreted as algebraic structures. This naturally hints at a question, which to my knowledge went unasked for a surprisingly long time: when does a logic admit a "nice algebraic interpretation?"<br />
<br />
Perhaps surprisingly, this is actually a question which can be made precise enough to treat with interesting results. I'll sketch what is probably the first serious result along these lines, due to Blok and Pigozzi, and then say a bit about where this aspect of algebraic logic has gone from there.<br />
<br />
=== March 16 - Spring break - No seminar ===<br />
<br />
'''Due to the cancellation of face-to-face instruction in UW-Madison through at least April 10, the seminar is suspended until further notice'''<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19222Graduate Logic Seminar2020-03-08T19:56:54Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
Title: Algebraic logic and algebraizable logics<br />
<br />
Abstract: Arguably the oldest theme in what we would recognize as "mathematical logic" is the algebraic interpretation of logic, the most famous example of this being the connection between (classical) propositional logic and Boolean algebras. But underlying the subject of algebraic logic is the implicit assumption that many logical systems are "satisfyingly" interpreted as algebraic structures. This naturally hints at a question, which to my knowledge went unasked for a surprisingly long time: when does a logic admit a "nice algebraic interpretation?"<br />
<br />
Perhaps surprisingly, this is actually a question which can be made precise enough to treat with interesting results. I'll sketch what is probably the first serious result along these lines, due to Blok and Pigozzi, and then say a bit about where this aspect of algebraic logic has gone from there.<br />
<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - TBD ===<br />
=== March 30 - TBD ===<br />
<br />
=== April 6 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19214Graduate Logic Seminar2020-03-06T19:17:42Z<p>Omer: /* March 9 - Noah Schweber */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
Title: Algebraic logic and algebraizable logics<br />
<br />
Abstract: Arguably the oldest theme in what we would recognize as "mathematical logic" is the algebraic interpretation of logic, the most famous example of this being the connection between (classical) propositional logic and Boolean algebras. But underlying the subject of algebraic logic is the implicit assumption that many logical systems are "satisfyingly" interpreted as algebraic structures. This naturally hints at a question, which to my knowledge went unasked for a surprisingly long time: when does a logic admit a "nice algebraic interpretation?"<br />
<br />
Perhaps surprisingly, this is actually a question which can be made precise enough to treat with interesting results. I'll sketch what is probably the first serious result along these lines, due to Blok and Pigozzi, and then say a bit about where this aspect of algebraic logic has gone from there.<br />
<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - TBD ===<br />
<br />
=== April 6 - TBD ===<br />
<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19193Graduate Logic Seminar2020-03-04T21:25:37Z<p>Omer: /* April 6 - Josiah Jacobsen-Grocott */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - TBD ===<br />
<br />
=== April 6 - TBD ===<br />
<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19192Graduate Logic Seminar2020-03-04T21:25:24Z<p>Omer: /* March 30 - Josiah Jacobsen-Grocott */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - TBD ===<br />
<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19191Graduate Logic Seminar2020-03-04T21:25:02Z<p>Omer: /* March 30 - TBD */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19181Graduate Logic Seminar2020-03-02T23:15:36Z<p>Omer: /* March 9 - Patrick Nicodemus */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Noah Schweber ===<br />
<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19163Graduate Logic Seminar2020-02-27T18:18:44Z<p>Omer: /* March 2 - Patrick Nicodemus */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
<br />
Title: A Sheaf-theoretic generalization of Los's theorem<br />
<br />
Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.<br />
<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19078Graduate Logic Seminar2020-02-20T19:52:11Z<p>Omer: /* February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.<br />
<br />
Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.<br />
<br />
'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19038Graduate Logic Seminar2020-02-17T02:01:26Z<p>Omer: /* February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
Title: A Characterization of Strongly $\eta$-Representable Degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=19033Graduate Logic Seminar2020-02-14T20:12:53Z<p>Omer: /* February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
<br />
Title: A characterization of strongly $\eta$-representable degrees.<br />
<br />
Abstract:<br />
$\eta$-representations are a way of coding sets in computable linear orders that were first<br />
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to<br />
characterize the sets with $\eta$-representations as well as the sets with subclasses of<br />
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only<br />
class where the order type of the representation is unique.<br />
<br />
We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$<br />
approximations. We use connected approximations to<br />
give a characterization of the degrees with strong $\eta$-representations as well new<br />
characterizations of the subclasses of $\eta$-representations with known characterizations.<br />
<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18963Graduate Logic Seminar2020-02-10T02:30:46Z<p>Omer: /* February 10 - No seminar - speaker was sick */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar (speaker was sick) ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18962Graduate Logic Seminar2020-02-10T02:30:30Z<p>Omer: /* February 10 - James Hanson */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - No seminar - speaker was sick ===<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18961Graduate Logic Seminar2020-02-10T02:30:03Z<p>Omer: /* February 17 - James Hanson */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 17 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18949Graduate Logic Seminar2020-02-06T19:26:57Z<p>Omer: /* February 10 - James Hanson */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - James Hanson ===<br />
<br />
Title: The Topology of Definable Sets in Continuous Logic<br />
<br />
Abstract: We will look at the topology of certain special subsets of type spaces in continuous logic, such as definable sets. In the process we will characterize those type spaces which have 'enough definable sets' and look at some counterexamples to things which would have been nice.<br />
<br />
=== February 17 - James Hanson ===<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18819Graduate Logic Seminar2020-01-29T15:46:58Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - James Hanson ===<br />
=== February 17 - James Hanson ===<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== March 30 - Josiah Jacobsen-Grocott ===<br />
=== April 6 - Josiah Jacobsen-Grocott ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18815Graduate Logic Seminar2020-01-28T17:08:55Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - James Hanson ===<br />
=== February 17 - James Hanson ===<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - TBD ===<br />
=== March 30 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== April 6 - TBD ===<br />
=== April 13 - Faculty at conference - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18763Graduate Logic Seminar2020-01-23T19:35:22Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - James Hanson ===<br />
=== February 17 - James Hanson ===<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - Spring break - No seminar ===<br />
=== March 23 - TBD ===<br />
=== March 30 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== April 6 - TBD ===<br />
=== April 13 - Passover - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18762Graduate Logic Seminar2020-01-23T19:19:27Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - Talk by visitor - No seminar ===<br />
=== February 10 - James Hanson ===<br />
=== February 17 - James Hanson ===<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - TBD ===<br />
=== March 23 - TBD ===<br />
=== March 30 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== April 6 - TBD ===<br />
=== April 13 - Passover - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18761Graduate Logic Seminar2020-01-23T18:46:07Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 21 - Talk by visitor - No seminar ===<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - James Hanson ===<br />
=== February 10 - James Hanson ===<br />
=== February 17 - TBD ===<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - TBD ===<br />
=== March 23 - TBD ===<br />
=== March 30 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== April 6 - TBD ===<br />
=== April 13 - Passover - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18760Graduate Logic Seminar2020-01-23T18:45:58Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 21 - Talk by visitor - No seminar ===<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - James Hanson ===<br />
=== February 10 - James Hanson ===<br />
=== February 17 - TBD ===<br />
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - TBD ===<br />
=== March 23 ===<br />
=== March 30 - Two short talks - Harry Main-Luu and Daniel Belin ===<br />
=== April 6 - TBD ===<br />
=== April 13 - Passover - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18759Graduate Logic Seminar2020-01-23T18:43:04Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 21 - Talk by visitor - No seminar ===<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - James Hanson ===<br />
=== February 10 - James Hanson ===<br />
=== February 17 - TBD ===<br />
=== February 24 - Two short talks ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 - Patrick Nicodemus ===<br />
=== March 16 - TBD ===<br />
=== March 23 - Two short talks ===<br />
=== March 30 - TBD ===<br />
=== April 6 - TBD ===<br />
=== April 13 - Passover - No seminar ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18750Graduate Logic Seminar2020-01-23T00:56:53Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
== Spring 2020 - Tentative schedule ==<br />
<br />
=== January 21 - Talk by visitor - No seminar ===<br />
=== January 28 - Talk by visitor - No seminar ===<br />
=== February 3 - James Hanson ===<br />
=== February 10 - James Hanson ===<br />
=== February 17 - Two short talks ===<br />
=== February 24 - Patrick Nicodemus ===<br />
=== March 2 - Patrick Nicodemus ===<br />
=== March 9 ===<br />
=== March 16 ===<br />
=== March 23 - Two short talks ===<br />
=== March 30 ===<br />
=== April 6 ===<br />
=== April 13 ===<br />
=== April 20 - Harry Main-Luu ===<br />
=== April 27 - Harry Main-Luu ===<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18749Graduate Logic Seminar2020-01-23T00:48:16Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18599Graduate Logic Seminar2020-01-07T16:14:38Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B215.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18407Graduate Logic Seminar2019-11-13T20:39:17Z<p>Omer: /* November 18 - Iván Ongay Valverde */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
Title: A couple of summer results<br />
<br />
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.<br />
<br />
In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18400Graduate Logic Seminar2019-11-12T00:17:30Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Iván Ongay Valverde ===<br />
<br />
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===<br />
<br />
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===<br />
<br />
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18381Graduate Logic Seminar2019-11-10T16:24:46Z<p>Omer: /* November 25 - No seminar */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - TBD ===<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18380Graduate Logic Seminar2019-11-10T16:08:00Z<p>Omer: /* November 25 - Two short talks */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - No seminar ===<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18296Graduate Logic Seminar2019-11-03T21:18:49Z<p>Omer: /* November 4 - Two short talks */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F<br />
<br />
A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18251Graduate Logic Seminar2019-10-25T15:52:54Z<p>Omer: /* November 4 - Two short talks */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - TBD<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18250Graduate Logic Seminar2019-10-25T15:52:42Z<p>Omer: /* November 11 - Manlio Valenti */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - TBD<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br/><br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18249Graduate Logic Seminar2019-10-25T15:51:29Z<p>Omer: /* November 11 - Manlio Valenti */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - TBD<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
<br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18248Graduate Logic Seminar2019-10-25T15:50:33Z<p>Omer: /* November 11 - Manlio Valenti I */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - TBD<br />
<br />
=== November 11 - Manlio Valenti ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18247Graduate Logic Seminar2019-10-25T15:50:16Z<p>Omer: /* November 11 - Manlio Valenti I */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - TBD<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
Title: The complexity of closed Salem sets (full length)<br />
<br />
Abstract:<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18246Graduate Logic Seminar2019-10-25T15:49:52Z<p>Omer: /* November 4 - Two short talks */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)<br />
<br />
A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets. <br />
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line. <br />
<br />
'''Patrick Nicodemus''' - TBD<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18245Graduate Logic Seminar2019-10-25T15:46:03Z<p>Omer: /* October 28 - Two short talks */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic<br />
<br />
The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
Manlio Valenti and Patrick Nicodemus<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18219Graduate Logic Seminar2019-10-20T16:29:38Z<p>Omer: /* October 23 - Tejas Bhojraj */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson'''<br />
<br />
TBA<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
Manlio Valenti and Patrick Nicodemus<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18218Graduate Logic Seminar2019-10-20T16:29:16Z<p>Omer: /* October 21 - Tejas Bhojraj */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 23 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson'''<br />
<br />
TBA<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
Manlio Valenti and Patrick Nicodemus<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18211Graduate Logic Seminar2019-10-18T14:57:27Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 21 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)<br />
<br />
In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):<br />
<br />
- Is the axiom weaker if we demand that $W$ is clopen?<br />
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?<br />
- Can we expand this axiom to spaces that are not second countable and metric?<br />
<br />
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.<br />
<br />
'''James Earnest Hanson'''<br />
<br />
TBA<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
Manlio Valenti and Patrick Nicodemus<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18194Graduate Logic Seminar2019-10-16T16:26:01Z<p>Omer: /* November 4 - Two short talks */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 21 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
Iván Ongay Valverde and James Earnest Hanson<br />
<br />
=== November 4 - Two short talks ===<br />
<br />
Manlio Valenti and Patrick Nicodemus<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18193Graduate Logic Seminar2019-10-16T04:45:14Z<p>Omer: /* Fall 2019 - Tentative schedule */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 21 - Tejas Bhojraj ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
Iván Ongay Valverde and James Earnest Hanson<br />
<br />
=== November 4 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Madison_Math_Circle&diff=18127Madison Math Circle2019-10-07T23:39:31Z<p>Omer: /* Meetings for Fall 2019 */</p>
<hr />
<div>[[Image:logo.png|right|440px]]<br />
<br />
For the site in Spanish, visit [[Math Circle de Madison]]<br />
=What is a Math Circle?=<br />
The Madison Math Circle is a weekly series of mathematically based activities aimed at interested middle school and high school students. It is an outreach program organized by the UW Math Department. Our goal is to provide a taste of exciting ideas in math and science. In the past we've had talks about plasma and weather in outer space, video game graphics, and encryption. In the sessions, students (and parents) are often asked to explore problems on their own, with the presenter facilitating a discussion. The talks are independent of one another, so new students are welcome at any point.<br />
<br />
The level of the audience varies quite widely, including a mix of middle school and high school students, and the speakers generally address this by considering subjects that will be interesting for a wide range of students.<br />
<br />
<br />
[[Image: MathCircle_2.jpg|500px]] [[Image: MathCircle_4.jpg|500px]] <br />
<br />
<br />
After each talk we'll have pizza provided by the Mathematics Department, and students will have an opportunity to mingle and chat with the speaker and with other participants, to ask questions about some of the topics that have been discussed, and also about college, careers in science, etc.<br />
<br />
'''The Madison Math circle was featured in Wisconsin State Journal:''' [http://host.madison.com/wsj/news/local/education/local_schools/school-spotlight-madison-math-circle-gives-young-students-a-taste/article_77f5c042-0b3d-11e1-ba5f-001cc4c03286.html check it out]!<br />
<br />
=All right, I want to come!=<br />
<br />
We have a weekly meeting, <b>Monday at 6pm in 3255 Helen C White Library</b>, during the school year. <b>New students are welcome at any point! </b> There is no fee and the talks are independent of one another, so you can just show up any week, but we ask all participants to take a moment to register by following the link below:<br />
<br />
[https://uwmadison.co1.qualtrics.com/jfe/form/SV_e9WdAs2SXNurWFD '''Math Circle Registration Form''']<br />
<br />
All of your information is kept private, and is only used by the Madison Math Circle organizer to help run the Circle. <br />
<br />
If you are a student, we hope you will tell other interested students about these talks, and speak with your parents or with your teacher about organizing a car pool to the UW campus. If you are a parent or a teacher, we hope you'll tell your students about these talks and organize a car pool to the UW (all talks take place in 3255 Helen C White Library, on the UW-Madison campus, right next to the Memorial Union).<br />
<br />
<br />
==Directions and parking==<br />
Our meetings are held on the 3rd floor of Helen C. White Hall in room 3255.<br />
<br />
<div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><br />
[[File: Helencwhitemap.png|400px]]</div><br />
<br />
'''Parking.''' Parking on campus is rather limited. Here is as list of some options:<br />
<br />
*There is a parking garage in the basement of Helen C. White, with an hourly rate. Enter from Park Street.<br />
*A 0.5 mile walk to Helen C. White Hall via [http://goo.gl/cxTzJY these directions], many spots ('''free starting 4:30pm''') [http://goo.gl/maps/Gkx1C in Lot 26 along Observatory Drive].<br />
*A 0.3 mile walk to Helen C. White Hall via [http://goo.gl/yMJIRd these directions], many spots ('''free starting 4:30pm''') [http://goo.gl/maps/vs17X in Lot 34]. <br />
*A 0.3 mile walk to Helen C. White Hall via [http://goo.gl/yMJIRd these directions], 2 metered spots (25 minute max) [http://goo.gl/maps/ukTcu in front of Lathrop Hall].<br />
*A 0.2 mile walk to Helen C. White Hall via [http://goo.gl/b8pdk2 these directions] 6 metered spots (25 minute max) around [http://goo.gl/maps/6EAnc the loop in front of Chadbourne Hall] .<br />
*For more information, see the [http://transportation.wisc.edu/parking/parking.aspx UW-Madison Parking Info website].<br />
<br />
==Email list==<br />
The best way to keep up to date with the what is going is by signing up for our email list. Send an empty email to join-mathcircle@lists.wisc.edu<br />
<br />
==Contact the organizers==<br />
The Madison Math Circle is organized by a group of professors and graduate students from the [http://www.math.wisc.edu Department of Mathematics] at the UW-Madison. If you have any questions, suggestions for topics, or so on, just email the '''organizers''' [mailto:cbooms@wisc.edu here]. We are always interested in feedback!<br />
<center><br />
<gallery widths=480px heights=240px mode="packed"><br />
File:de.jpg|[https://www.math.wisc.edu/~derman/ Prof. Daniel Erman]<br />
<!--File:Betsy.jpg|[http://www.math.wisc.edu/~stovall/ Prof. Betsy Stovall]--><br />
</gallery><br />
<br />
<gallery widths=500px heights=250px mode="packed"><br />
<!--File:juliettebruce.jpg|[http://www.math.wisc.edu/~juliettebruce/ Juliette Bruce] File:Ee.jpg|[http://www.math.wisc.edu/~evaelduque/ Eva Elduque] File:mrjulian.jpg|[http://www.math.wisc.edu/~mrjulian/ Ryan Julian] File:soumyasankar.jpg|[http://www.math.wisc.edu/~soumyasankar Soumya Sankar]--><br />
File:caitlynbooms.jpg|[https://sites.google.com/wisc.edu/cbooms Caitlyn Booms]<br />
File:colincrowley.jpg|[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
File:hyunjongkim.jpg|Hyun Jong Kim<br />
File:connorsimpson.jpg|[http://www.math.wisc.edu/~csimpson6/ Connor Simpson]<br />
</gallery><br />
</center><br />
<br />
==Donations==<br />
Please consider donating to the Madison Math Circle. As noted in our [https://www.math.wisc.edu/wiki/images/Math_Circle_Newsletter.pdf annual report], our main costs consist of pizza and occasional supplies for the speakers. So far our costs have been covered by donations from the UW Mathematics Department as well as a generous gifts from a private donor. But our costs are rising, primarily because this year we expect to hold more meetings than in any previous year. In fact, this year, we expect to spend at least $2500 on pizza and supplies alone.<br />
<br />
So please consider donating to support your math circle! The easiest way to donate is to go to the link:<br />
<br />
[http://www.math.wisc.edu/donate Online Donation Link]<br />
<br />
There are instructions on that page for donating to the Math Department. <b> Be sure and add a Gift Note saying that the donation is intended for the "Madison Math Circle"!</b> The money goes into the Mathematics Department Annual Fund and is routed through the University of Wisconsin Foundation, which is convenient for record-keeping, etc.<br />
<br />
Alternately, you can bring a check to one of the Math Circle Meetings. If you write a check, be sure to make it payable to the "WFAA" and add the note "Math Circle Donation" on the check. <br />
<br />
Or you can just pay in cash, and we'll give you a receipt.<br />
<br />
==Help us grow!==<br />
If you like Math Circle, please help us continue to grow! Students, parents, and teachers can help by:<br />
*Posting our [https://www.math.wisc.edu/wiki/images/MMC_Flyer_2016.pdf '''flyer'''] at schools or anywhere that might have interested students<br />
*Discussing the Math Circle with students, parents, teachers, administrators, and others<br />
*Making an announcement about Math Circle at PTO meetings<br />
*Donating to Math Circle<br />
Contact the organizers if you have questions or your own ideas about how to help out.<br />
<br />
=Meetings for Fall 2019=<br />
<br />
<center><br />
<br />
Talks start at '''6pm in room 3255 of Helen C. White Library''', unless otherwise noted.<br />
<br />
</center><br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="1" cellpadding="14" cellspacing="0"<br />
|-<br />
! colspan="3" style="background: #e8b2b2;" align="center" | Fall 2019<br />
|-<br />
! Date !! Speaker !! Topic<br />
|-<br />
| September 23, 2019 || Soumya Sankar || Why don't map makers like high heels?<br />
|-<br />
| September 30, 2019 || Erika Pirnes || Why do ice hockey players fall in love with mathematicians?<br />
|-<br />
| October 7, 2019 || Uri Andrews || Self-reference, proofs, and computer programming<br />
|-<br />
| October 14, 2019 || James Hanson || When is a puzzle impossible?<br />
|-<br />
| October 21, 2019 || Owen Goff || TBD<br />
|-<br />
| October 28, 2019 || TBD || TBD<br />
|-<br />
| November 4, 2019 || Omer Mermelstein || TBD<br />
|-<br />
| November 11, 2019 || Colin Crowley || TBD<br />
|-<br />
| November 18, 2019 || Daniel Corey || TBD<br />
|-<br />
|}<br />
<br />
</center><br />
<br />
=Off-Site Meetings=<br />
<br />
We will hold some Math Circle meetings at local high schools on early release days. If you are interesting in having us come to your high school, please contact us!<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="1" cellpadding="14" cellspacing="0"<br />
|-<br />
! colspan="5" style="background: #e8b2b2;" align="center" | Fall 2019<br />
|-<br />
|-<br />
! Date !! Location !! Speaker !! Title !! Abstract<br />
|-<br />
| October 7, 2019 || 2:45pm East High || Solly Parenti || Tangled Up in Two || Every tangled cord you have ever encountered is secretly a number. Once you learn how to count these cords, cleaning your room will be as easy as 1-2-3.<br />
|-<br />
| November 4, 2019 || 2:45pm James Madison Memorial || Daniel Erman || TBD ||<br />
|-<br />
| November 11, 2019 || 2:45pm East High || Maya Banks || TBD ||<br />
|-<br />
| December 16, 2019 || 2:45pm James Madison Memorial || Caitlyn Booms || TBD ||<br />
|}<br />
</center><br />
<br />
=Useful Resources=<br />
==Annual Reports==<br />
[https://www.math.wisc.edu/wiki/images/Math_Circle_Newsletter.pdf 2013-2014 Annual Report]<br />
<br />
== Archived Abstracts ==<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_2016-2017 2016 - 2017 Math Circle Page]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts_2016-2017 2016 - 2017 Abstracts]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_2015-2016 2015 - 2016 Math Circle Page]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Math_Circle_de_Madison_2015-2016 2015 - 2016 Math Circle Page (Spanish)]<br />
<br />
[https://www.math.wisc.edu/wiki/index.php/Madison_Math_Circle_Abstracts_2015-2016 2015 - 2015 Abstracts]<br />
<br />
[[Archived Math Circle Material]]<br />
<br />
==Link for presenters (in progress)==<br />
[https://www.math.wisc.edu/wiki/index.php/Math_Circle_Presentations Advice For Math Circle Presenters]<br />
<br />
[http://www.mathcircles.org/math-problems-2/ Sample Talk Ideas/Problems]<br />
<br />
[http://www.mathcircles.org/content/circle-box "Circle in a Box"]</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18097Graduate Logic Seminar2019-10-03T16:05:24Z<p>Omer: </p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj I ===<br />
<br />
Title: Solovay and Schnorr randomness for infinite sequences of qubits.<br />
<br />
Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.<br />
<br />
=== October 21 - Tejas Bhojraj II ===<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
Iván Ongay Valverde and James Earnest Hanson<br />
<br />
=== November 4 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18067Graduate Logic Seminar2019-10-01T17:33:25Z<p>Omer: /* October 21 - Tejas Bhojraj II - Date may change */</p>
<hr />
<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
<br />
<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj I ===<br />
<br />
=== October 21 - Tejas Bhojraj II ===<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
Iván Ongay Valverde and James Earnest Hanson<br />
<br />
=== November 4 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omerhttps://www.math.wisc.edu/wiki/index.php?title=Graduate_Logic_Seminar&diff=18066Graduate Logic Seminar2019-10-01T17:33:16Z<p>Omer: /* October 14 - Tejas Bhojraj I - Date may change */</p>
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<div>The Graduate Logic Seminar is an informal space where graduate student and professors present topics related to logic which are not necessarly original or completed work. This is a space focused principally on practicing presentation skills or learning materials that are not usually presented in a class.<br />
<br />
* '''When:''' Mondays 4p-5p<br />
* '''Where:''' Van Vleck B223.<br />
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]<br />
<br />
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.<br />
<br />
Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu<br />
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<br />
<br />
== Fall 2019 - Tentative schedule ==<br />
<br />
=== September 5 - Organizational meeting ===<br />
<br />
=== September 9 - No seminar ===<br />
<br />
=== September 16 - Daniel Belin ===<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic<br />
<br />
Abstract: Lachlan, in a result later refined and clarified by Odifreddi, proved in 1970 that initial segments of the m-degrees can be embedded as an upper semilattice formed as the limit of finite distributive lattices. This allows us to show that the many-one degrees codes satisfiability in second-order arithmetic, due to a later result of Nerode and Shore. We will take a journey through Lachlan's rather complicated construction which sheds a great deal of light on the order-theoretic properties of many-one reducibility.<br />
<br />
=== September 23 - Daniel Belin ===<br />
<br />
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued<br />
<br />
=== September 30 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Models<br />
<br />
Abstract: Infinatary logic extends the notions of first order logic by allowing infinite formulas. Scott's Isomorphism Theorem states that any countable structure can be characterized up to isomorphism by a single countable sentence. Closely related to the complexity of this sentence is what is known as the Scott Rank of the structure. In this talk we restrict our attention to computable models and look at an upper bound on the Scott Rank of such structures.<br />
<br />
=== October 7 - Josiah Jacobsen-Grocott ===<br />
<br />
Title: Scott Rank of Computable Codels - Continued<br />
<br />
=== October 14 - Tejas Bhojraj I ===<br />
<br />
=== October 21 - Tejas Bhojraj II - Date may change ===<br />
<br />
=== October 28 - Two short talks ===<br />
<br />
Iván Ongay Valverde and James Earnest Hanson<br />
<br />
=== November 4 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== November 11 - Manlio Valenti I ===<br />
<br />
=== November 18 - Manlio Valenti II ===<br />
<br />
=== November 25 - Two short talks ===<br />
Speakers TBD<br />
<br />
=== December 2 - Iván Ongay Valverde I ===<br />
<br />
=== December 9 - Iván Ongay Valverde II ===<br />
<br />
==Previous Years==<br />
<br />
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].</div>Omer