Difference between revisions of "Graduate Logic Seminar"

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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.
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The Graduate Logic Seminar is an informal space where graduate students 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.
  
* '''When:''' Mondays 4p-5p
+
* '''When:''' TBA
* '''Where:''' Van Vleck B215.
+
* '''Where:''' on line (ask for code).
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]
+
* '''Organizers:''' [https://www.math.wisc.edu/~jgoh/ Jun Le Goh]
  
 
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.
 
The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.
Line 9: Line 9:
 
Sign up for the graduate logic seminar mailing list:  join-grad-logic-sem@lists.wisc.edu
 
Sign up for the graduate logic seminar mailing list:  join-grad-logic-sem@lists.wisc.edu
  
 +
== Spring 2021 - Tentative schedule ==
  
 +
=== February 16 3:30PM - Short talk by Sarah Reitzes (University of Chicago) ===
  
== Fall 2019 - Tentative schedule ==
+
Title: Reduction games over $\mathrm{RCA}_0$
  
=== September 5 - Organizational meeting ===
+
Abstract: In this talk, I will discuss joint work with Damir D. Dzhafarov and Denis R. Hirschfeldt. Our work centers on the characterization of problems P and Q such that P $\leq_{\omega}$ Q, as well as problems P and Q such that $\mathrm{RCA}_0 \vdash$ Q $\to$ P, in terms of winning strategies in certain games. These characterizations were originally introduced by Hirschfeldt and Jockusch. I will discuss extensions and generalizations of these characterizations, including a certain notion of compactness that allows us, for strategies satisfying particular conditions, to bound the number of moves it takes to win. This bound is independent of the instance of the problem P being considered. This allows us to develop the idea of Weihrauch and generalized Weihrauch reduction over some base theory. Here, we will focus on the base theory $\mathrm{RCA}_0$. In this talk, I will explore these notions of reduction among various principles, including bounding and induction principles.
  
=== September 9 - No seminar ===
+
=== March 23 4:15PM - Steffen Lempp ===
  
=== September 16 - Daniel Belin ===
+
Title: Degree structures and their finite substructures
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic
 
  
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.
+
Abstract: Many problems in mathematics can be viewed as being coded by sets of natural numbers (as indices).
 +
One can then define the relative computability of sets of natural numbers in various ways, each leading to a precise notion of “degree” of a problem (or set).
 +
In each case, these degrees form partial orders, which can be studied as algebraic structures.
 +
The study of their finite substructures leads to a better understanding of the partial order as a whole.
  
=== September 23 - Daniel Belin ===
+
=== March 30 4PM - Alice Vidrine ===
  
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued
+
Title: Categorical logic for realizability, part I: Categories and the Yoneda Lemma
  
=== September 30 - Josiah Jacobsen-Grocott ===
+
Abstract: An interesting strand of modern research on realizability--a semantics for non-classical logic based on a notion of computation--uses the language of toposes and Grothendieck fibrations to study mathematical universes whose internal notion of truth is similarly structured by computation. The purpose of this talk is to establish the basic notions of category theory required to understand the tools of categorical logic developed in the sequel, with the end goal of understanding the realizability toposes developed by Hyland, Johnstone, and Pitts. The talk will cover the definitions of category, functor, natural transformation, adjunctions, and limits/colimits, with a heavy emphasis on the ubiquitous notion of representability.
  
Title: Scott Rank of Computable Models
+
[https://hilbert.math.wisc.edu/wiki/images/Cat-slides-1.pdf Link to slides]
 
 
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.
 
 
 
=== October 7 - Josiah Jacobsen-Grocott ===
 
 
 
Title: Scott Rank of Computable Codels - Continued
 
 
 
=== October 14 - Tejas Bhojraj ===
 
 
 
Title: Solovay and Schnorr randomness for infinite sequences of qubits.
 
 
 
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.
 
 
 
=== October 23 - Tejas Bhojraj ===
 
 
 
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued
 
 
 
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.
 
 
 
=== October 28 - Two short talks ===
 
 
 
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)
 
 
 
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):
 
 
 
- Is the axiom weaker if we demand that $W$ is clopen?
 
- If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$?
 
- Can we expand this axiom to spaces that are not second countable and metric?
 
 
 
These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.
 
 
 
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic
 
 
 
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.
 
 
 
=== November 4 - Two short talks ===
 
 
 
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)
 
 
 
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.
 
 
 
'''Patrick Nicodemus''' - Proof theory of Second Order Arithmetic and System F
 
 
 
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.
 
 
 
=== November 11 - Manlio Valenti ===
 
 
 
Title: The complexity of closed Salem sets (full length)
 
 
 
Abstract:
 
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.
 
 
 
=== November 18 - Iván Ongay Valverde ===
 
 
 
Title: A couple of summer results
 
 
 
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.
 
 
 
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.
 
 
 
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===
 
 
 
=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===
 
 
 
=== December 9 - Anniversary of the death of Pope Pius IV - No seminar  ===
 
  
 
==Previous Years==
 
==Previous Years==
  
 
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].
 
The schedule of talks from past semesters can be found [[Graduate Logic Seminar, previous semesters|here]].

Latest revision as of 17:19, 30 March 2021

The Graduate Logic Seminar is an informal space where graduate students 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.

  • When: TBA
  • Where: on line (ask for code).
  • Organizers: Jun Le Goh

The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact one of the organizers.

Sign up for the graduate logic seminar mailing list: join-grad-logic-sem@lists.wisc.edu

Spring 2021 - Tentative schedule

February 16 3:30PM - Short talk by Sarah Reitzes (University of Chicago)

Title: Reduction games over $\mathrm{RCA}_0$

Abstract: In this talk, I will discuss joint work with Damir D. Dzhafarov and Denis R. Hirschfeldt. Our work centers on the characterization of problems P and Q such that P $\leq_{\omega}$ Q, as well as problems P and Q such that $\mathrm{RCA}_0 \vdash$ Q $\to$ P, in terms of winning strategies in certain games. These characterizations were originally introduced by Hirschfeldt and Jockusch. I will discuss extensions and generalizations of these characterizations, including a certain notion of compactness that allows us, for strategies satisfying particular conditions, to bound the number of moves it takes to win. This bound is independent of the instance of the problem P being considered. This allows us to develop the idea of Weihrauch and generalized Weihrauch reduction over some base theory. Here, we will focus on the base theory $\mathrm{RCA}_0$. In this talk, I will explore these notions of reduction among various principles, including bounding and induction principles.

March 23 4:15PM - Steffen Lempp

Title: Degree structures and their finite substructures

Abstract: Many problems in mathematics can be viewed as being coded by sets of natural numbers (as indices). One can then define the relative computability of sets of natural numbers in various ways, each leading to a precise notion of “degree” of a problem (or set). In each case, these degrees form partial orders, which can be studied as algebraic structures. The study of their finite substructures leads to a better understanding of the partial order as a whole.

March 30 4PM - Alice Vidrine

Title: Categorical logic for realizability, part I: Categories and the Yoneda Lemma

Abstract: An interesting strand of modern research on realizability--a semantics for non-classical logic based on a notion of computation--uses the language of toposes and Grothendieck fibrations to study mathematical universes whose internal notion of truth is similarly structured by computation. The purpose of this talk is to establish the basic notions of category theory required to understand the tools of categorical logic developed in the sequel, with the end goal of understanding the realizability toposes developed by Hyland, Johnstone, and Pitts. The talk will cover the definitions of category, functor, natural transformation, adjunctions, and limits/colimits, with a heavy emphasis on the ubiquitous notion of representability.

Link to slides

Previous Years

The schedule of talks from past semesters can be found here.