Difference between revisions of "Graduate Logic Seminar"
Line 13: | Line 13: | ||
This day we decided the schedule for the semester. | This day we decided the schedule for the semester. | ||
− | === February 5, Uri Andrews === | + | === February 5, [http://www.math.wisc.edu/~andrews/ Uri Andrews] === |
Title: Building Models of Strongly Minimal Theories - Part 1 | Title: Building Models of Strongly Minimal Theories - Part 1 | ||
− | Abstract: Since I'm talking in the Tuesday seminar as well, I'll use | + | 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 |
− | the Monday seminar talk to do some background on the topic and some | + | 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 |
− | lemmas that will go into the proofs in Tuesday's talk. There will be | + | the general topic of answering the following question: What do you need to know about a strongly minimal theory in order to compute |
− | (I hope) some theorems of interest to see on both days, and both on | + | copies of all of its countable models. I'll start with a definition for strongly minimal theories and build up from there. |
− | the general topic of answering the following question: What do you | ||
− | need to know about a strongly minimal theory in order to compute | ||
− | copies of all of its countable models. I'll start with a definition | ||
− | for strongly minimal theories and build up from there. | ||
=== February 12, James Hanson === | === February 12, James Hanson === | ||
Line 30: | Line 26: | ||
Title: Finding Definable Sets in Continuous Logic | Title: Finding Definable Sets in Continuous Logic | ||
− | Abstract: In order to be useful the notion of a 'definable set' in | + | Abstract: In order to be useful the notion of a 'definable set' in continuous logic is stricter than a naive comparison to discrete logic |
− | continuous logic is stricter than a naive comparison to discrete logic | + | would suggest. As a consequence, even in relatively tame theories there can be very few definable sets. For example, there is a |
− | would suggest. As a consequence, even in relatively tame theories | + | superstable theory with no non-trivial definable sets. As we'll see, however, there are many definable sets in omega-stable, |
− | there can be very few definable sets. For example, there is a | ||
− | superstable theory with no non-trivial definable sets. As we'll see, | ||
− | however, there are many definable sets in omega-stable, | ||
omega-categorical, and other small theories. | omega-categorical, and other small theories. | ||
− | === February 19, Noah Schweber === | + | === February 19, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] === |
Title: Proper forcing | Title: Proper forcing | ||
− | Abstract: Although a given forcing notion may have nice properties on | + | Abstract: Although a given forcing notion may have nice properties on its own, those properties might vanish when we apply it repeatedly. |
− | its own, those properties might vanish when we apply it repeatedly. | + | Early preservation results (that is, theorems saying that the iteration of forcings with a nice property retains that nice property) |
− | Early preservation results (that is, theorems saying that the | + | were fairly limited, and things really got off the ground with Shelah's invention of "proper forcing." Roughly speaking, a forcing is |
− | iteration of forcings with a nice property retains that nice property) | + | proper if it can be approximated by elementary submodels of the universe in a particularly nice way. I'll define proper forcing and |
− | were fairly limited, and things really got off the ground with | ||
− | Shelah's invention of "proper forcing." Roughly speaking, a forcing is | ||
− | proper if it can be approximated by elementary submodels of the | ||
− | universe in a particularly nice way. I'll define proper forcing and | ||
sketch some applications. | sketch some applications. | ||
Line 56: | Line 45: | ||
Title: A survey of computable and constructive mathematics in economic history | Title: A survey of computable and constructive mathematics in economic history | ||
− | === March 5, Tamvana Makulumi === | + | === March 5, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] === |
Title: Convexly Orderable Groups | Title: Convexly Orderable Groups | ||
− | === March 12, Dan Turetsky (University of Notre Dame) === | + | === March 12, [https://math.nd.edu/people/visiting-faculty/daniel-turetsky/ Dan Turetsky] (University of Notre Dame) === |
Title: Structural Jump | Title: Structural Jump | ||
− | === March 19, Ethan McCarthy === | + | === March 19, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] === |
Title: Networks and degrees of points in non-second countable spaces | Title: Networks and degrees of points in non-second countable spaces | ||
Line 72: | Line 61: | ||
Title: Characterizing Finite Nilpotent Groups via Word Maps | Title: Characterizing Finite Nilpotent Groups via Word Maps | ||
− | Abstract: In this talk, we will examine a novel characterization of finite | + | 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. |
− | 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. | ||
=== April 9, Tejas Bhojraj === | === April 9, Tejas Bhojraj === | ||
Line 81: | Line 67: | ||
Title: Quantum Randomness | Title: Quantum Randomness | ||
− | Abstract: I will read the paper by Nies and Scholz where they define a notion of | + | 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. |
− | 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. | ||
=== April 16, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde] === | === April 16, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde] === | ||
Line 94: | Line 77: | ||
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. | 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. | ||
− | This is a work in progress, so this talk will have multiple open questions and opportunities for feedback and public participation (hopefully). | + | This is a work in progress, so this talk will have multiple open questions and opportunities for feedback and public participation.(hopefully). |
=== April 23, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] (Thesis Defense) === | === April 23, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] (Thesis Defense) === | ||
Line 103: | Line 86: | ||
=== April 30, [http://www.math.uconn.edu/~westrick/ Linda Brown Westrick] (from University Of Connecticut) === | === April 30, [http://www.math.uconn.edu/~westrick/ Linda Brown Westrick] (from University Of Connecticut) === | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
Title: TBA | Title: TBA | ||
Line 120: | Line 97: | ||
This day we decided the schedule for the semester. | This day we decided the schedule for the semester. | ||
− | === September 18, | + | === September 18, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] === |
− | |||
− | |||
− | |||
− | |||
− | + | Title: The Kunen inconsistency | |
− | + | Abstract: While early large cardinal axioms were usually defined combinatorially - e.g., cardinals satisfying a version of Ramsey's | |
+ | theorem - later focus shifted to model-theoretic definitions, specifically definitions in terms of elementary embeddings of the | ||
+ | whole universe of sets. At the lowest level, a measurable cardinal is one which is the least cardinal moved (= critical point) by a | ||
+ | nontrivial elementary embedding from V into some inner model M. | ||
− | + | There are several variations on this theme yielding stronger and stronger large cardinal notions; one of the most important is the | |
+ | inclusion of *correctness properties* of the target model M. The strongest such correctness property is total correctness: M=V. The | ||
+ | critical point of an elementary embedding from V to V is called a *Reinhardt cardinal*. Shortly after their introduction in Reinhardt's | ||
+ | thesis, however, the existence of a Reinhardt cardinal was shown to be inconsistent with ZFC. | ||
− | + | I'll present this argument, and talk a bit about the role of choice. | |
− | + | === September 25, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] === | |
− | + | Title: Hindman's theorem via ultrafilters | |
− | + | 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. | |
− | + | === October 2, James Hanson === | |
− | + | Title: The Gromov-Hausdorff metric on type space in continuous logic | |
− | + | 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. | |
− | + | === October 9, James Hanson === | |
− | + | Title: Morley rank and stability in continuous logic | |
− | + | 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 | |
+ | > 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. | ||
− | + | === October 23, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] === | |
− | + | Title: Boxy sets in ordered convexly-orderable structures. | |
− | === October 30, Iván Ongay-Valverde === | + | === October 30, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde] === |
− | Title: | + | Title: Dancing SCCA and other Coloring Axioms |
− | Abstract: | + | 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. |
− | === November 6, | + | === November 6, Wil Cocke === |
− | Title: | + | Title: Two new characterizations of nilpotent groups |
− | Abstract: | + | 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. |
− | + | Or... | |
− | Title: | + | Title: Centralizing Propagating Properties of Groups |
− | Abstract: | + | 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. |
− | === November | + | === November 13, [https://www.math.wisc.edu/~lempp/ Steffen Lempp] === |
− | Title: | + | Title: The computational complexity of properties of finitely presented groups |
− | Abstract: | + | Abstract: I will survey index set complexity results on finitely presented groups. |
− | === November | + | === November 20, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] === |
− | Title: | + | Title: Strong Difference Randomness |
− | Abstract: | + | 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. |
− | === December 4, | + | === December 4, Tejas Bhojraj === |
− | Title: | + | Title: Quantum Algorithmic Randomness |
− | Abstract: | + | 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. |
− | === December 11, | + | === December 11, Grigory Terlov === |
− | Title: | + | Title: The Logic of Erdős–Rényi Graphs |
− | |||
− | |||
==Previous Years== | ==Previous Years== | ||
The schedule of talks from past semesters can be found [[Logic Graduate Seminar, previous semesters|here]]. | The schedule of talks from past semesters can be found [[Logic Graduate Seminar, previous semesters|here]]. |
Revision as of 18:34, 18 April 2018
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 an space focus principally in practicing presentation skills or learning materials that are not usually presented on a class.
- When: Mondays, 4:00 PM – 5:00 PM (unless otherwise announced).
- Where: Van Vleck B235 (unless otherwise announced).
- Organizers: Mariya Soskava
Talks schedule are arrange and decide at the beginning of each semester. If you would like to participate, please contact one of the organizers.
Contents
- 1 Spring 2018
- 1.1 January 29, Organizational meeting
- 1.2 February 5, Uri Andrews
- 1.3 February 12, James Hanson
- 1.4 February 19, Noah Schweber
- 1.5 February 26, Patrick Nicodemus
- 1.6 March 5, Tamvana Makulumi
- 1.7 March 12, Dan Turetsky (University of Notre Dame)
- 1.8 March 19, Ethan McCarthy
- 1.9 April 2, Wil Cocke
- 1.10 April 9, Tejas Bhojraj
- 1.11 April 16, Iván Ongay-Valverde
- 1.12 April 23, Ethan McCarthy (Thesis Defense)
- 1.13 April 30, Linda Brown Westrick (from University Of Connecticut)
- 2 Fall 2017
- 2.1 September 11, Organizational meeting
- 2.2 September 18, Noah Schweber
- 2.3 September 25, Noah Schweber
- 2.4 October 2, James Hanson
- 2.5 October 9, James Hanson
- 2.6 October 23, Tamvana Makulumi
- 2.7 October 30, Iván Ongay-Valverde
- 2.8 November 6, Wil Cocke
- 2.9 November 13, Steffen Lempp
- 2.10 November 20, Ethan McCarthy
- 2.11 December 4, Tejas Bhojraj
- 2.12 December 11, Grigory Terlov
- 3 Previous Years
Spring 2018
January 29, Organizational meeting
This day we decided the schedule for the semester.
February 5, Uri Andrews
Title: Building Models of Strongly Minimal Theories - Part 1
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 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 the general topic of answering the following question: What do you need to know about a strongly minimal theory in order to compute copies of all of its countable models. I'll start with a definition for strongly minimal theories and build up from there.
February 12, James Hanson
Title: Finding Definable Sets in Continuous Logic
Abstract: In order to be useful the notion of a 'definable set' in continuous logic is stricter than a naive comparison to discrete logic would suggest. As a consequence, even in relatively tame theories there can be very few definable sets. For example, there is a superstable theory with no non-trivial definable sets. As we'll see, however, there are many definable sets in omega-stable, omega-categorical, and other small theories.
February 19, Noah Schweber
Title: Proper forcing
Abstract: Although a given forcing notion may have nice properties on its own, those properties might vanish when we apply it repeatedly. Early preservation results (that is, theorems saying that the iteration of forcings with a nice property retains that nice property) were fairly limited, and things really got off the ground with Shelah's invention of "proper forcing." Roughly speaking, a forcing is proper if it can be approximated by elementary submodels of the universe in a particularly nice way. I'll define proper forcing and sketch some applications.
February 26, Patrick Nicodemus
Title: A survey of computable and constructive mathematics in economic history
March 5, Tamvana Makulumi
Title: Convexly Orderable Groups
March 12, Dan Turetsky (University of Notre Dame)
Title: Structural Jump
March 19, Ethan McCarthy
Title: Networks and degrees of points in non-second countable spaces
April 2, Wil Cocke
Title: Characterizing Finite Nilpotent Groups via Word Maps
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.
April 9, Tejas Bhojraj
Title: Quantum Randomness
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.
April 16, Iván Ongay-Valverde
Title: What can we say about sets made by the union of Turing equivalence classes?
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?
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.
This is a work in progress, so this talk will have multiple open questions and opportunities for feedback and public participation.(hopefully).
April 23, Ethan McCarthy (Thesis Defense)
Title: TBA
Abstract: TBA
April 30, Linda Brown Westrick (from University Of Connecticut)
Title: TBA
Abstract: TBA
Fall 2017
September 11, Organizational meeting
This day we decided the schedule for the semester.
September 18, Noah Schweber
Title: The Kunen inconsistency
Abstract: While early large cardinal axioms were usually defined combinatorially - e.g., cardinals satisfying a version of Ramsey's theorem - later focus shifted to model-theoretic definitions, specifically definitions in terms of elementary embeddings of the whole universe of sets. At the lowest level, a measurable cardinal is one which is the least cardinal moved (= critical point) by a nontrivial elementary embedding from V into some inner model M.
There are several variations on this theme yielding stronger and stronger large cardinal notions; one of the most important is the inclusion of *correctness properties* of the target model M. The strongest such correctness property is total correctness: M=V. The critical point of an elementary embedding from V to V is called a *Reinhardt cardinal*. Shortly after their introduction in Reinhardt's thesis, however, the existence of a Reinhardt cardinal was shown to be inconsistent with ZFC.
I'll present this argument, and talk a bit about the role of choice.
September 25, Noah Schweber
Title: Hindman's theorem via ultrafilters
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.
October 2, James Hanson
Title: The Gromov-Hausdorff metric on type space in continuous logic
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.
October 9, James Hanson
Title: Morley rank and stability in continuous logic
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 > 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.
October 23, Tamvana Makulumi
Title: Boxy sets in ordered convexly-orderable structures.
October 30, Iván Ongay-Valverde
Title: Dancing SCCA and other Coloring Axioms
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.
November 6, Wil Cocke
Title: Two new characterizations of nilpotent groups
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.
Or...
Title: Centralizing Propagating Properties of Groups
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.
November 13, Steffen Lempp
Title: The computational complexity of properties of finitely presented groups
Abstract: I will survey index set complexity results on finitely presented groups.
November 20, Ethan McCarthy
Title: Strong Difference Randomness
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.
December 4, Tejas Bhojraj
Title: Quantum Algorithmic Randomness
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.
December 11, Grigory Terlov
Title: The Logic of Erdős–Rényi Graphs
Previous Years
The schedule of talks from past semesters can be found here.