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.
+
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:''' Mondays 3:30-4:30 PM
* '''Where:''' Van Vleck B215.
+
* '''Where:''' Van Vleck B139
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]
+
* '''Organizers:''' Karthik Ravishankar and [https://sites.google.com/wisc.edu/antonio Antonio Nakid Cordero]
  
 
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 2020 - Tentative schedule ==
+
== Fall 2022 ==
  
=== January 28 - Talk by visitor - No seminar ===
+
=== September 12 - Organizational Meeting ===
=== February 3 - Talk by visitor - No seminar ===
 
=== February 10 - James Hanson ===
 
  
Title: The Topology of Definable Sets in Continuous Logic
+
We will meet to assign speakers to dates.
  
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.
+
=== '''September 19 - Karthik Ravishankar''' ===
 +
'''Title:''' Lowness for Isomorphism ([https://wiki.math.wisc.edu/images/Karthik_talk.pdf Slides])
  
=== February 17 - James Hanson ===
+
'''Abstract:''' A Turing degree is said to be low for isomorphism if it can only compute an isomorphism between computable structures only when a computable isomorphism already exists. In this talk, we show that the measure of the class of low for isomorphism sets in Cantor space is 0 and that no Martin Lof random is low for isomorphism.
  
Title: The Topology of Definable Sets in Continuous Logic
+
=== '''September 26 - Antonio Nakid Cordero'''  ===
 +
'''Title:''' When Models became Polish: an introduction to the Topological Vaught Conjecture ([[Media:GradLogSem - Topological Vaught Conjecture.pdf|Slides]])
  
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.
+
'''Abstract:''' Vaught's Conjecture, originally asked by Vaught in 1961, is one of the most (in)famous open problems in mathematical logic. The conjecture is that a complete theory on a countable language must either have countably-many or continuum-many non-isomorphic models. In this talk, we will discuss some of the main ideas that surround this conjecture, with special emphasis on a topological generalization in terms of the continuous actions of Polish groups.
  
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===
+
=== '''October 10 - Yunting Zhang''' ===
=== March 2 - Patrick Nicodemus ===
+
'''Title:''' Some History of Logic ([[Media:Godel.pdf|Slides]])
=== March 9 - Patrick Nicodemus ===
 
=== March 16 - Spring break - No seminar ===
 
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===
 
=== March 30 - Josiah Jacobsen-Grocott ===
 
=== April 6 - Josiah Jacobsen-Grocott ===
 
=== April 13 - Faculty at conference - No seminar ===
 
=== April 20 - Harry Main-Luu ===
 
=== April 27 - Harry Main-Luu ===
 
  
 +
'''Abstract:''' The lives of great thinkers are sometimes overshadowed by their achievements-and there is perhaps no better illustration of this phenomenon than the life and work of Gödel. Take a look at Gödel's own timeline and see how wars and other mathematicians influenced him.
  
 +
=== '''October 17 - Alice Vidrine''' ===
 +
'''Title:''' Local operators, bilayer Turing reducibility, and enumeration Weihrauch degrees ([[Media:LTeW-talk.pdf|Slides]])
  
== Fall 2019 ==
+
'''Abstract:'''  Realizability toposes have a rich variety of subtoposes, corresponding to their local operators. These local operators are somewhat difficult to study in their usual form, which seems far removed from the usual objects of computability theoretic study. Recent work by Takayuki Kihara has given a characterization of the local operators on the effective topos in computability theoretic terms related to Weihrauch reduction, and which generalizes to several other realizability toposes of possible interest to computability theorists. This narrative-focused talk outlines what a realizability topos looks like, what local operators are, what Kihara's bilayer Turing reduction looks like, and how this leads to preliminary questions about a relative of the Weihrauch degrees based on enumeration reduction.
  
=== September 5 - Organizational meeting ===
+
=== '''October 24 - Hongyu Zhu''' ===
 +
'''Title:''' Investigating Natural Theories through the Consistency Operator ([[Media:ConsistencyOperator.pdf|Slides]])
  
=== September 9 - No seminar ===
+
'''Abstract:''' The phenomenon that "natural" theories tend to be linearly ordered in terms of consistency strength is a long-standing mystery. One approach to solving the problem is Martin's Conjecture, which roughly claims that the only natural functions on the Turing degrees are transfinite iterates of the Turing jump. In this talk we will focus on a similar approach, working inside the Lindenbaum algebra of elementary arithmetic instead of the Turing degrees. Here, the consistency operator takes the role of the jump. We will see that while some nice analogous claims can be established, there are also counterexamples that prevent us from strengthening the results in various ways.
  
=== September 16 - Daniel Belin ===
+
=== '''October 31 - Break for Halloween''' ===
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.
+
=== '''November 7 - John Spoerl''' ===
 +
'''Title:''' Universal Algorithms
  
=== September 23 - Daniel Belin ===
+
'''Abstract:''' Among the many ways we can flex Gödel’s Incompleteness theorems, there is one that feels especially strange: there is a partial computable function F, such that F gives no output in the standard model of arithmetic, but for any function G on natural numbers (computable or not) there is a non-standard model in which F behaves exactly as G. I’ll discuss some related arguments and some philosophical questions this may raise about our notions of finiteness and determinism.
  
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued
+
=== '''November 14 - Josiah Jacobsen-Grocott''' ===
 +
'''Title:''' The Paris-Harrington Theorem
  
=== September 30 - Josiah Jacobsen-Grocott ===
+
'''Abstract:''' In this talk, I will present the proof of the Paris-Harrington theorem. The
 +
Paris-Harrington theorem states that over PA the consistency of Peano arithmetic is equivalent to
 +
a strengthening of the finite Ramsey theorem. This was the first example of a result from "ordinary
 +
mathematics" that can not be proven by PA. The aim of this talk is to cover the main logical
 +
steps in this proof and give some of the combinatorics.
  
Title: Scott Rank of Computable Models
+
=== '''November 21 - Karthik Ravishankar''' ===
 +
'''Title:''' The computing power of Baire space vs Cantor Space
  
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.
+
'''Abstract:''' Generic Muchnik reducibility extends the notion of Muchnik reducibility to the uncountable setting. In this talk we'll look at some recent work which comes up with another technique of constructing a structure of degree strictly between Cantor space and Baire Space. We'll see that there is a generic copy of Cantor space which computes no escaping function while every copy of Baire space computes a dominating function. We then construct a structure of intermediate degree which always computes an escaping function but no dominating function.
  
=== October 7 - Josiah Jacobsen-Grocott ===
+
=== '''November 28 - Logan Heath''' ===
 +
'''Title''': A Mathematical Analysis of Theories of Generative Grammar: The Peters-Ritchie Theorem
  
Title: Scott Rank of Computable Codels - Continued
+
'''Abstract''': This is a preview of a talk intended primarily for undergraduate students in linguistics who have just completed a first course in syntax. The goal of the talk is to spark interest in applications of mathematical techniques to the study of theories of syntax. We will focus on the Peters-Ritchie Theorem which shows that transformational grammars can generate languages which are c.e., but not computable.
  
=== October 14 - Tejas Bhojraj ===
+
=== '''December 5 - Logan Heath''' ===
  
Title: Solovay and Schnorr randomness for infinite sequences of qubits.
+
=== '''December 12 - Yuxiao Fu''' ===
  
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.
+
== Previous Years ==
 
 
=== 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==
 
  
 
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 23:42, 27 November 2022

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 3:30-4:30 PM
  • Where: Van Vleck B139
  • Organizers: Karthik Ravishankar and Antonio Nakid Cordero

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

Fall 2022

September 12 - Organizational Meeting

We will meet to assign speakers to dates.

September 19 - Karthik Ravishankar

Title: Lowness for Isomorphism (Slides)

Abstract: A Turing degree is said to be low for isomorphism if it can only compute an isomorphism between computable structures only when a computable isomorphism already exists. In this talk, we show that the measure of the class of low for isomorphism sets in Cantor space is 0 and that no Martin Lof random is low for isomorphism.

September 26 - Antonio Nakid Cordero

Title: When Models became Polish: an introduction to the Topological Vaught Conjecture (Slides)

Abstract: Vaught's Conjecture, originally asked by Vaught in 1961, is one of the most (in)famous open problems in mathematical logic. The conjecture is that a complete theory on a countable language must either have countably-many or continuum-many non-isomorphic models. In this talk, we will discuss some of the main ideas that surround this conjecture, with special emphasis on a topological generalization in terms of the continuous actions of Polish groups.

October 10 - Yunting Zhang

Title: Some History of Logic (Slides)

Abstract: The lives of great thinkers are sometimes overshadowed by their achievements-and there is perhaps no better illustration of this phenomenon than the life and work of Gödel. Take a look at Gödel's own timeline and see how wars and other mathematicians influenced him.

October 17 - Alice Vidrine

Title: Local operators, bilayer Turing reducibility, and enumeration Weihrauch degrees (Slides)

Abstract: Realizability toposes have a rich variety of subtoposes, corresponding to their local operators. These local operators are somewhat difficult to study in their usual form, which seems far removed from the usual objects of computability theoretic study. Recent work by Takayuki Kihara has given a characterization of the local operators on the effective topos in computability theoretic terms related to Weihrauch reduction, and which generalizes to several other realizability toposes of possible interest to computability theorists. This narrative-focused talk outlines what a realizability topos looks like, what local operators are, what Kihara's bilayer Turing reduction looks like, and how this leads to preliminary questions about a relative of the Weihrauch degrees based on enumeration reduction.

October 24 - Hongyu Zhu

Title: Investigating Natural Theories through the Consistency Operator (Slides)

Abstract: The phenomenon that "natural" theories tend to be linearly ordered in terms of consistency strength is a long-standing mystery. One approach to solving the problem is Martin's Conjecture, which roughly claims that the only natural functions on the Turing degrees are transfinite iterates of the Turing jump. In this talk we will focus on a similar approach, working inside the Lindenbaum algebra of elementary arithmetic instead of the Turing degrees. Here, the consistency operator takes the role of the jump. We will see that while some nice analogous claims can be established, there are also counterexamples that prevent us from strengthening the results in various ways.

October 31 - Break for Halloween

November 7 - John Spoerl

Title: Universal Algorithms

Abstract: Among the many ways we can flex Gödel’s Incompleteness theorems, there is one that feels especially strange: there is a partial computable function F, such that F gives no output in the standard model of arithmetic, but for any function G on natural numbers (computable or not) there is a non-standard model in which F behaves exactly as G. I’ll discuss some related arguments and some philosophical questions this may raise about our notions of finiteness and determinism.

November 14 - Josiah Jacobsen-Grocott

Title: The Paris-Harrington Theorem

Abstract: In this talk, I will present the proof of the Paris-Harrington theorem. The Paris-Harrington theorem states that over PA the consistency of Peano arithmetic is equivalent to a strengthening of the finite Ramsey theorem. This was the first example of a result from "ordinary mathematics" that can not be proven by PA. The aim of this talk is to cover the main logical steps in this proof and give some of the combinatorics.

November 21 - Karthik Ravishankar

Title: The computing power of Baire space vs Cantor Space

Abstract: Generic Muchnik reducibility extends the notion of Muchnik reducibility to the uncountable setting. In this talk we'll look at some recent work which comes up with another technique of constructing a structure of degree strictly between Cantor space and Baire Space. We'll see that there is a generic copy of Cantor space which computes no escaping function while every copy of Baire space computes a dominating function. We then construct a structure of intermediate degree which always computes an escaping function but no dominating function.

November 28 - Logan Heath

Title: A Mathematical Analysis of Theories of Generative Grammar: The Peters-Ritchie Theorem

Abstract: This is a preview of a talk intended primarily for undergraduate students in linguistics who have just completed a first course in syntax. The goal of the talk is to spark interest in applications of mathematical techniques to the study of theories of syntax. We will focus on the Peters-Ritchie Theorem which shows that transformational grammars can generate languages which are c.e., but not computable.

December 5 - Logan Heath

December 12 - Yuxiao Fu

Previous Years

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