Difference between revisions of "Graduate Logic Seminar"

From UW-Math Wiki
Jump to navigation Jump to search
(80 intermediate revisions by 4 users not shown)
Line 1: Line 1:
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:''' Tuesdays 4-5 PM
* '''Where:''' Van Vleck B215.
+
* '''Where:''' Van Vleck 901
* '''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 2020 - Tentative schedule ==
+
== Spring 2022 ==
  
=== January 28 - Talk by visitor - No seminar ===
+
The graduate logic seminar this semester will be run as MATH 975. Please enroll if you wish to participate.
=== February 3 - Talk by visitor - No seminar ===
 
=== February 10 - James Hanson ===
 
=== February 17 - James Hanson ===
 
=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===
 
=== March 2 - Patrick Nicodemus ===
 
=== March 9 - Patrick Nicodemus ===
 
=== March 16 - Spring break - No seminar ===
 
=== March 23 - TBD ===
 
=== March 30 - Two short talks - Harry Main-Luu and Daniel Belin ===
 
=== April 6 - TBD ===
 
=== April 13 - Passover - No seminar ===
 
=== April 20 - Harry Main-Luu ===
 
=== April 27 - Harry Main-Luu ===
 
  
 +
We plan to cover the first 9 parts of [https://blog.nus.edu.sg/matwong/teach/modelarith/ Tin Lok Wong's notes], as well as a few other relevant topics which are not covered in the notes:
 +
* Properness of the induction/bounding hierarchy (chapter 10 of Models of Peano Arithmetic by Kaye is a good source)
 +
* Tennenbaum's theorem (this is a quick consequence of the main theorem of part 4, so it should be combined with part 4 or part 5)
 +
* Other facts found in chapter 1 of [http://homepages.math.uic.edu/~marker/marker-thesis.pdf David Marker's thesis].
  
 +
=== January 25 - organizational meeting ===
  
== Fall 2019 ==
+
We will meet to assign speakers to dates.
  
=== September 5 - Organizational meeting ===
+
=== February 1 - Steffen Lempp ===
  
=== September 9 - No seminar ===
+
I will give an overview of the topics we will cover:
  
=== September 16 - Daniel Belin ===
+
1. the base theory PA^- and the induction and bounding axioms for Sigma_n-formulas, and how they relate to each other,
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.
+
2. the equivalence of Sigma_n-induction with a version of Sigma_n-separation (proved by H. Friedman),
  
=== September 23 - Daniel Belin ===
+
3. the Grzegorczyk hierarchy of fast-growing functions,
  
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued
+
4. end extensions and cofinal extensions,
  
=== September 30 - Josiah Jacobsen-Grocott ===
+
5. recursive saturation and resplendency,
  
Title: Scott Rank of Computable Models
+
6. standard systems and coded types,
  
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.
+
7. the McDowell-Specker Theorem that every model of PA has a proper elementary end extension, and
  
=== October 7 - Josiah Jacobsen-Grocott ===
+
8. Gaifman's theorem that every model of PA has a minimal elementary end extension.
  
Title: Scott Rank of Computable Codels - Continued
+
I will sketch the basic definitions and state the main theorems, in a form that one can appreciate without too much
 +
background.
  
=== October 14 - Tejas Bhojraj ===
+
=== February 8 - Karthik Ravishankar ===
  
Title: Solovay and Schnorr randomness for infinite sequences of qubits.
+
Title: Collection axioms
  
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.
+
We will discuss parts 1 and 2 of Wong's notes.
  
=== October 23 - Tejas Bhojraj ===
+
=== February 15 - Karthik Ravishankar, Yunting Zhang ===
  
Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued
+
Title: Collection axioms/The Weak König Lemma
  
Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.
+
Karthik will finish part 2 of Wong's notes. Then Yunting will start on part 3 of Wong's notes.
  
=== October 28 - Two short talks ===
+
=== February 22 - Yunting Zhang ===
  
'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)
+
Title: The Weak König Lemma
  
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):
+
We will finish part 3 of Wong's notes.
  
- Is the axiom weaker if we demand that $W$ is clopen?
+
=== March 22 - Ang Li ===
- 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.
+
Title: The Arithmetized Completeness Theorem
  
'''James Earnest Hanson''' - Strongly minimal sets in continuous logic
+
We will discuss part 4 of Wong's notes.
  
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.
+
=== March 29 - Ang Li ===
  
=== November 4 - Two short talks ===
+
Title: The Arithmetized Completeness Theorem
  
'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)
+
We will finish part 4 of Wong's notes.
  
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.
+
=== April 5 - Antonio Nákid Cordero ===
<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
+
Title: Semiregular cuts
  
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.
+
We will start on part 5 of Wong's notes.
  
=== November 11 - Manlio Valenti ===
+
=== April 12 - Antonio Nákid Cordero/Alice Vidrine ===
  
Title: The complexity of closed Salem sets (full length)
+
Title: Semiregular cuts/End and cofinal extensions
  
Abstract:
+
We will finish part 5 of Wong's notes and then start on part 6.
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 ===
+
=== April 19 - Alice Vidrine ===
  
Title: A couple of summer results
+
Title: End and cofinal extensions
  
Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.
+
We will finish part 6 of Wong's notes.
  
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.
+
=== May 3 - No seminar today ===
  
=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===
+
== Previous Years ==
 
 
=== 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]].

Revision as of 11:44, 2 May 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: Tuesdays 4-5 PM
  • Where: Van Vleck 901
  • 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 2022

The graduate logic seminar this semester will be run as MATH 975. Please enroll if you wish to participate.

We plan to cover the first 9 parts of Tin Lok Wong's notes, as well as a few other relevant topics which are not covered in the notes:

  • Properness of the induction/bounding hierarchy (chapter 10 of Models of Peano Arithmetic by Kaye is a good source)
  • Tennenbaum's theorem (this is a quick consequence of the main theorem of part 4, so it should be combined with part 4 or part 5)
  • Other facts found in chapter 1 of David Marker's thesis.

January 25 - organizational meeting

We will meet to assign speakers to dates.

February 1 - Steffen Lempp

I will give an overview of the topics we will cover:

1. the base theory PA^- and the induction and bounding axioms for Sigma_n-formulas, and how they relate to each other,

2. the equivalence of Sigma_n-induction with a version of Sigma_n-separation (proved by H. Friedman),

3. the Grzegorczyk hierarchy of fast-growing functions,

4. end extensions and cofinal extensions,

5. recursive saturation and resplendency,

6. standard systems and coded types,

7. the McDowell-Specker Theorem that every model of PA has a proper elementary end extension, and

8. Gaifman's theorem that every model of PA has a minimal elementary end extension.

I will sketch the basic definitions and state the main theorems, in a form that one can appreciate without too much background.

February 8 - Karthik Ravishankar

Title: Collection axioms

We will discuss parts 1 and 2 of Wong's notes.

February 15 - Karthik Ravishankar, Yunting Zhang

Title: Collection axioms/The Weak König Lemma

Karthik will finish part 2 of Wong's notes. Then Yunting will start on part 3 of Wong's notes.

February 22 - Yunting Zhang

Title: The Weak König Lemma

We will finish part 3 of Wong's notes.

March 22 - Ang Li

Title: The Arithmetized Completeness Theorem

We will discuss part 4 of Wong's notes.

March 29 - Ang Li

Title: The Arithmetized Completeness Theorem

We will finish part 4 of Wong's notes.

April 5 - Antonio Nákid Cordero

Title: Semiregular cuts

We will start on part 5 of Wong's notes.

April 12 - Antonio Nákid Cordero/Alice Vidrine

Title: Semiregular cuts/End and cofinal extensions

We will finish part 5 of Wong's notes and then start on part 6.

April 19 - Alice Vidrine

Title: End and cofinal extensions

We will finish part 6 of Wong's notes.

May 3 - No seminar today

Previous Years

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