Graduate Logic Seminar: Difference between revisions

From UW-Math Wiki
Jump to navigation Jump to search
No edit summary
(64 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 - No seminar - speaker was sick ===


=== February 17 - James Hanson ===
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].


Title: The Topology of Definable Sets in Continuous Logic
=== January 25 - organizational meeting ===


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.
We will meet to assign speakers to dates.


=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===
=== February 1 - Steffen Lempp ===
=== March 2 - Patrick Nicodemus ===
=== 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 ===


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,


== Fall 2019 ==
2. the equivalence of Sigma_n-induction with a version of Sigma_n-separation (proved by H. Friedman),


=== September 5 - Organizational meeting ===
3. the Grzegorczyk hierarchy of fast-growing functions,


=== September 9 - No seminar ===
4. end extensions and cofinal extensions,


=== September 16 - Daniel Belin ===
5. recursive saturation and resplendency,
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.
6. standard systems and coded types,


=== September 23 - Daniel Belin ===
7. the McDowell-Specker Theorem that every model of PA has a proper elementary end extension, and


Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued
8. Gaifman's theorem that every model of PA has a minimal elementary end extension.


=== September 30 - Josiah Jacobsen-Grocott ===
I will sketch the basic definitions and state the main theorems, in a form that one can appreciate without too much
background.


Title: Scott Rank of Computable Models
=== February 8 - Karthik Ravishankar ===


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.
Title: Collection axioms


=== October 7 - Josiah Jacobsen-Grocott ===
We will discuss parts 1 and 2 of Wong's notes.


Title: Scott Rank of Computable Codels - Continued
== Previous Years ==
 
=== 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==


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 00:06, 26 January 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.

Previous Years

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