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
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* '''When:''' Tuesdays 4-5 PM
* '''Where:''' Van Vleck B223.
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* '''Where:''' Van Vleck 901
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]
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* '''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.
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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
  
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== Spring 2022 ==
  
 +
The graduate logic seminar this semester will be run as MATH 975. Please enroll if you wish to participate.
  
== Fall 2019 - Tentative schedule ==
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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].
  
=== September 5 - Organizational meeting ===
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=== January 25 - organizational meeting ===
  
=== September 9 - No seminar ===
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We will meet to assign speakers to dates.
  
=== September 16 - Daniel Belin ===
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=== February 1 - Steffen Lempp ===
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.
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I will give an overview of the topics we will cover:
  
=== September 23 - Daniel Belin ===
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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 - Continued
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2. the equivalence of Sigma_n-induction with a version of Sigma_n-separation (proved by H. Friedman),
  
=== September 30 - Josiah Jacobsen-Grocott ===
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3. the Grzegorczyk hierarchy of fast-growing functions,
  
Title: Scott Rank of Computable Models
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4. end extensions and cofinal extensions,
  
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.
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5. recursive saturation and resplendency,
  
=== October 7 - Josiah Jacobsen-Grocott ===
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6. standard systems and coded types,
  
Title: Scott Rank of Computable Codels - Continued
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7. the McDowell-Specker Theorem that every model of PA has a proper elementary end extension, and
  
=== October 14 - Tejas Bhojraj ===
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8. Gaifman's theorem that every model of PA has a minimal elementary end extension.
  
Title: Solovay and Schnorr randomness for infinite sequences of qubits.
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I will sketch the basic definitions and state the main theorems, in a form that one can appreciate without too much
 +
background.
  
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.
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== 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.
 
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.
 
 
 
'''Patrick Nicodemus''' - TBD
 
 
 
=== 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.
 
 
 
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.
 
 
 
=== November 18 - Manlio Valenti II ===
 
 
 
=== November 25 - Two short talks ===
 
Speakers TBD
 
 
 
=== December 2 - Iván Ongay Valverde I ===
 
 
 
=== December 9 - Iván Ongay Valverde II ===
 
 
 
==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 16:28, 18 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.

Previous Years

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