Difference between revisions of "Graduate Logic Seminar"

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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.
 
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
+
* '''When:''' Tuesdays 4-5 PM
* '''Where:''' on line (ask for code).
+
* '''Where:''' Van Vleck 901
 
* '''Organizers:''' [https://www.math.wisc.edu/~jgoh/ Jun Le Goh]
 
* '''Organizers:''' [https://www.math.wisc.edu/~jgoh/ Jun Le Goh]
  
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
  
== Fall 2021 - Tentative schedule ==
+
== Spring 2022 ==
  
To see what's happening in the Logic qual preparation sessions click [[Logic Qual Prep|here]].
+
The graduate logic seminar this semester will be run as MATH 975. Please enroll if you wish to participate.
  
== Spring 2021 - Tentative schedule ==
+
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].
  
=== February 16 3:30PM - Short talk by Sarah Reitzes (University of Chicago) ===
+
=== January 25 - organizational meeting ===
  
Title: Reduction games over $\mathrm{RCA}_0$
+
We will meet to assign speakers to dates.
  
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.
+
=== February 1 - Steffen Lempp ===
  
=== March 23 4:15PM - Steffen Lempp ===
+
I will give an overview of the topics we will cover:  
  
Title: Degree structures and their finite substructures
+
1. the base theory PA^- and the induction and bounding axioms for Sigma_n-formulas, and how they relate to each other,
  
Abstract: Many problems in mathematics can be viewed as being coded by sets of natural numbers (as indices).
+
2. the equivalence of Sigma_n-induction with a version of Sigma_n-separation (proved by H. Friedman),
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 ===
+
3. the Grzegorczyk hierarchy of fast-growing functions,
  
Title: Categorical logic for realizability, part I: Categories and the Yoneda Lemma
+
4. end extensions and cofinal extensions,
  
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.
+
5. recursive saturation and resplendency,
  
[https://hilbert.math.wisc.edu/wiki/images/Cat-slides-1.pdf Link to slides]
+
6. standard systems and coded types,
  
=== April 27 4PM - Alice Vidrine ===
+
7. the McDowell-Specker Theorem that every model of PA has a proper elementary end extension, and
  
Title: Categorical logic for realizability, part II
+
8. Gaifman's theorem that every model of PA has a minimal elementary end extension.
  
Abstract: Realizability is an approach to semantics for non-classical logic that interprets propositions by sets of abstract computational data. One modern approach to realizability makes heavy use of the notion of a topos, a type of category that behaves like a universe of non-standard sets. In preparation for introducing realizability toposes, the present talk will be a brisk introduction to the notion of a topos, with an emphasis on their logical aspects. In particular, we will look at the notion of a subobject classifier and the internal logic to which it gives rise.
+
I will sketch the basic definitions and state the main theorems, in a form that one can appreciate without too much
 +
background.
  
==Previous Years==
+
=== 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 [[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 12:44, 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.