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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 B223.
* '''Where:''' Van Vleck B223
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]
* '''Organizers:''' [https://uriandrews.netlify.app/ Uri Andrews] and [https://sites.google.com/view/hongyu-zhu/ Hongyu Zhu]


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.


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:  [mailto:join-grad-logic-sem@lists.wisc.edu join-grad-logic-sem@lists.wisc.edu]


== Fall 2023 ==


The seminar will be run as a 1-credit seminar Math 975 in Fall 2023. If you are not enrolled but would like to audit it, please contact [mailto:andrews@math.wisc.edu Uri Andrews] and [mailto:hongyu@math.wisc.edu Hongyu Zhu].


== Fall 2019 - Tentative schedule ==
While you are welcome (and encouraged) to present on a topic of your own choice, feel free to ask for help from faculties and/or other graduate students.


=== September 5 - Organizational meeting ===
Presentation Schedule: https://docs.google.com/spreadsheets/d/15Qd4EzrrKpn1Ct5tur1P_FDc2czsdAVnUf_pfp65Lb4/edit?usp=sharing


=== September 9 - No seminar ===
Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)


=== September 16 - Daniel Belin ===
Possible readings:
Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic
* (Elementary) Proof Theory: Chapters 4-7 of <i>[https://projecteuclid.org/ebooks/lecture-notes-in-logic/Aspects-of-Incompleteness/toc/lnl/1235416274 Aspects of Incompleteness]</i> by Per Lindström.
* An invitation to model-theoretic Galois theory.  <i>[https://arxiv.org/abs/0909.4340 On arxiv here.]</i>
* Variations on the Feferman-Vaught Theorem <i>[https://arxiv.org/abs/1812.02905 On arxiv here.]</i>
* Any of several papers on "Turing Computable Embeddings"
* Computability/Model/Set Theory: Consult faculties/students for recommended texts on specific areas.


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.
=== September 11 - Organizational Meeting ===


=== September 23 - Daniel Belin ===
We will meet to assign speakers to dates.


Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued
=== '''September 18 - Taeyoung Em''' ===
'''Title:''' Explicit construction of non-quasidetermined game on <math>\mathcal P(2^{\mathbb N})</math> without using A.C. ([https://wiki.math.wisc.edu/images/Gale-Stewart_implies_A.C..pdf Supplement])


=== September 30 - Josiah Jacobsen-Grocott ===
'''Abstract:''' We will go over briefly some basic information about trees and infinite games. Then we prove the Gale-Stewart Theorem. The proof of the theorem motivates definition of quasistrategy. Then we will briefly introduce Borel determinacy. We will go over how the usage of A.C. makes convenient for us to make a non-quasidetermined or undertermined game. We will give an explicit construction of a non-quasidetermined game on <math>\mathcal P(2^{\mathbb N})</math> without using A.C.


Title: Scott Rank of Computable Models
=== '''September 25 - Karthik Ravishankar''' ===
'''Title:''' Spectra of structures


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:''' One way to measure the complexity of a structure is via its spectrum - the set of Turing degrees of its copies. In this talk, we'll look at the definition and first properties of the spectrum followed by some examples. In particular, we'll show that the non-computable degrees and the hyperimmune degrees form a spectrum while the DNC degrees do not.


=== October 7 - Josiah Jacobsen-Grocott ===
=== '''October 2 - Hongyu Zhu''' ===
'''Title:''' Continuum Hypothesis: On Platonism and Pluralism ([https://wiki.math.wisc.edu/images/CH.pdf Slides] and [https://uwmadison.zoom.us/rec/share/lSe2BL28988PGmmthWKA6FM7bWOJ0eR6vxP4laS7O6ImNN2gQ5skPJ6-C8KlbGcm.G48mQQ0qlW-lo3gr Recording]; Passcode: .iXJs?1t)


Title: Scott Rank of Computable Codels - Continued
'''Abstract:''' Despite its independence from ZFC, the continuum hypothesis continues to be of interest to logicians. In this talk, we will see arguments for settling the truth of CH in one way or another (or yet another). We will see how mathematical arguments (the inner model program) are intertwined with philosophical beliefs (mathematical Platonism and pluralism) about the set-theoretic universe(s).


=== October 14 - Tejas Bhojraj ===
=== '''October 9 - Hannah Ashbach ''' ===
'''Title:''' An Introduction to Constructive Mathematics ([https://wiki.math.wisc.edu/images/An_Introduction_to_Constructive_Mathematics.pdf Slides] and [https://uwmadison.zoom.us/rec/share/UIxI2m2WWMnitZTd1oQxYxl6vpashJdvf_2UtN6erlFXN9yNgm7q0mhXw1min0-L.cz3b7tsbBPlsgxD6 Recording]; Passcode: 5.$c5L+2)


Title: Solovay and Schnorr randomness for infinite sequences of qubits.
'''Abstract:''' Have you ever written a mathematical proof and felt dissatisfied after writing QED? Perhaps you had proven the existence of a particularly complicated mathematical object, but you have no clue how that object may actually look or be constructed. Or perhaps you are ready to denounce the Axiom of Choice after reading about its far-reaching consequences. Constructive logic is a formal logic system that seeks to clear up these concerns for mathematicians, though not all mathematicians agree with the power it holds-- or takes away.


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 16 - Rune Chen ''' ===
'''Title:''' An Introduction to Model-Theoretic Galois Theory


=== October 23 - Tejas Bhojraj ===
'''Abstract:''' Given an arbitrary first-order theory with elimination of imaginaries, the standard Galois theory can be translated into a model-theoretic version, which can be further generalized and thus be useful in other algebraic settings. In this talk, we will give a brief introduction to model-theoretic Galois theory and then look at a simple case of model-theoretic Galois correspondence. We will skip the detailed discussion of elimination of imaginaries by choosing our theory T to be a first-order theory coding finite sets and working in a large enough saturated model M of T.


Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued
=== '''October 23 - John Spoerl ''' ===
'''Title:''' The Computational Content of Forcing


Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.
'''Abstract:''' Given information about a ground model, how much can you know about it’s forcing extensions? Given a forcing extension, how much can you know about its ground model? This talk will be a primer on forcing and a review of the main results of Hamkins, Miller and Williams’ 2020 paper: “Forcing as a Computational Process”.


=== October 28 - Two short talks ===
=== '''October 30 - Chiara Travesset ''' ===
'''Title:''' Mission-time LTL (MLTL) Formula Validation Via Regular Expressions


'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)
'''Abstract:''' Mission-time Linear Temporal Logic (mLTL) is an extension of propositional logic that includes temporal operators over finite intervals of time. A computation is a (finite) sequence consisting of a truth assignment for each propositional variable at each time step. This presentation will describe how we can use regular expressions to describe the structure of the computations that make a given mLTL formula true. We prove soundness and completeness, and also give an implemented algorithm (the WEST program) and analyze its complexity both theoretically and experimentally. We generate a test suite using control flow diagrams to robustly test the code. Finally, we present the REST theorem, which significantly simplifies certain sets of computations. This talk only requires basic familiarity with propositional logic and computer science.


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):
=== '''November 6 - Antonio Nakid Cordero ''' ===
'''Title:''' Martin's Conjecture in the Turing degrees and the Enumeration Degrees


- Is the axiom weaker if we demand that $W$ is clopen?
'''Abstract:''' Martin's Conjecture is an attempt to formalize the empirical phenomenon that "naturally occurring" Turing degrees are well-ordered. In this talk, I'll go over how one can make that into a precise mathematical statement, a uniform version of the conjecture that has been fully resolved, and some of the complications of trying to propose a similar conjecture in the enumeration degrees. If time allows, I'll report on my ongoing project in this direction.
- 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.
=== '''November 13 - Alice Vidrine ''' ===
'''Title:''' Enumeration Weihrauch degrees and closed choice problems ([https://wiki.math.wisc.edu/images/EWdeg_Grad_seminar_Fall_23.pdf Slides] and [https://uwmadison.zoom.us/rec/share/F_pDYkWhHGiMTy7Rkqrb_PpeJnjxBCo0ZIMOEHV4CP-XfJodbV-P8igStgVwiXq4.A7ONs9u5Em6-3Gvp Recording] Passcode: C89k#5&0)


'''James Earnest Hanson''' - Strongly minimal sets in continuous logic
'''Abstract:''' In the Weihrauch degrees, we think of elements of Baire space as names for elements of mathematical objects. In the enumeration Weihrauch (eW) degrees we follow a similar idea, but our names may only carry positive information. It turns out that this coding by positive information separates several choice problems that are equivalent in the Weihrauch degrees, and in this talk we look at some of the more striking examples and their proofs. We close with some discussion of various directions for further research.


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 20 - Logan Heath ''' ===
'''Title:''' A Proof Theorist's Guide to First Aid: CUT-Elimination and Other Results of Classical Proof Theory ([https://wiki.math.wisc.edu/images/A_Proof_Theorist%27s_Guide_to_First_Aid.pdf Slides])


=== November 4 - Two short talks ===
'''Abstract:''' When are correct proofs also "bad" proofs, how can we avoid them, and should we even try? In this talk we examine some early results of classical proof theory regarding the consistency of the sequent calculus for FOL and '''PA'''. In both cases our approach will be to avoid proofs which are technically correct, but also defective in some sense.


'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)
=== November 27 - Thanksgiving Break ===
=== '''December 4 - Mei Rose Connor ''' ===
'''Title:''' When ‘And’ is Too Much and ‘Or’ is Not Enough


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.
'''Abstract:''' This work was motivated by the seven–valued predication system of the Jaina religion, which states that no statement exists as an absolute. This system, known as ''Saptabhangivada'' in Sanskrit, has 7 truth values that are the non-empty subsets of the set {asti, nāsti, avaktavyah}. These words translate to: ''it is'', ''it is not'', and ''it is unassertible'', respectively. This talk will explore how these seven truth values relate to each other, the problems encountered when trying to formalise this system, and the author’s current resolution of said problems using lattice theory and introducing an eighth truth value.
<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
=== '''December 11 - Ang Li ''' ===
'''Title:''' TBA


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.
'''Abstract:''' TBA


=== November 11 - Manlio Valenti ===
<!-- Template


Title: The complexity of closed Salem sets (full length)
=== '''September 18 - Karthik Ravishankar''' ===
'''Title:''' Lowness for Isomorphism ([https://wiki.math.wisc.edu/images/Karthik_talk.pdf Slides])


Abstract:
'''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.
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 - Manlio Valenti II ===
-->


=== November 25 - No seminar ===
== Previous Years ==
 
=== 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 03:18, 27 November 2023

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 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 2023

The seminar will be run as a 1-credit seminar Math 975 in Fall 2023. If you are not enrolled but would like to audit it, please contact Uri Andrews and Hongyu Zhu.

While you are welcome (and encouraged) to present on a topic of your own choice, feel free to ask for help from faculties and/or other graduate students.

Presentation Schedule: https://docs.google.com/spreadsheets/d/15Qd4EzrrKpn1Ct5tur1P_FDc2czsdAVnUf_pfp65Lb4/edit?usp=sharing

Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)

Possible readings:

  • (Elementary) Proof Theory: Chapters 4-7 of Aspects of Incompleteness by Per Lindström.
  • An invitation to model-theoretic Galois theory. On arxiv here.
  • Variations on the Feferman-Vaught Theorem On arxiv here.
  • Any of several papers on "Turing Computable Embeddings"
  • Computability/Model/Set Theory: Consult faculties/students for recommended texts on specific areas.

September 11 - Organizational Meeting

We will meet to assign speakers to dates.

September 18 - Taeyoung Em

Title: Explicit construction of non-quasidetermined game on [math]\displaystyle{ \mathcal P(2^{\mathbb N}) }[/math] without using A.C. (Supplement)

Abstract: We will go over briefly some basic information about trees and infinite games. Then we prove the Gale-Stewart Theorem. The proof of the theorem motivates definition of quasistrategy. Then we will briefly introduce Borel determinacy. We will go over how the usage of A.C. makes convenient for us to make a non-quasidetermined or undertermined game. We will give an explicit construction of a non-quasidetermined game on [math]\displaystyle{ \mathcal P(2^{\mathbb N}) }[/math] without using A.C.

September 25 - Karthik Ravishankar

Title: Spectra of structures

Abstract: One way to measure the complexity of a structure is via its spectrum - the set of Turing degrees of its copies. In this talk, we'll look at the definition and first properties of the spectrum followed by some examples. In particular, we'll show that the non-computable degrees and the hyperimmune degrees form a spectrum while the DNC degrees do not.

October 2 - Hongyu Zhu

Title: Continuum Hypothesis: On Platonism and Pluralism (Slides and Recording; Passcode: .iXJs?1t)

Abstract: Despite its independence from ZFC, the continuum hypothesis continues to be of interest to logicians. In this talk, we will see arguments for settling the truth of CH in one way or another (or yet another). We will see how mathematical arguments (the inner model program) are intertwined with philosophical beliefs (mathematical Platonism and pluralism) about the set-theoretic universe(s).

October 9 - Hannah Ashbach

Title: An Introduction to Constructive Mathematics (Slides and Recording; Passcode: 5.$c5L+2)

Abstract: Have you ever written a mathematical proof and felt dissatisfied after writing QED? Perhaps you had proven the existence of a particularly complicated mathematical object, but you have no clue how that object may actually look or be constructed. Or perhaps you are ready to denounce the Axiom of Choice after reading about its far-reaching consequences. Constructive logic is a formal logic system that seeks to clear up these concerns for mathematicians, though not all mathematicians agree with the power it holds-- or takes away.

October 16 - Rune Chen

Title: An Introduction to Model-Theoretic Galois Theory

Abstract: Given an arbitrary first-order theory with elimination of imaginaries, the standard Galois theory can be translated into a model-theoretic version, which can be further generalized and thus be useful in other algebraic settings. In this talk, we will give a brief introduction to model-theoretic Galois theory and then look at a simple case of model-theoretic Galois correspondence. We will skip the detailed discussion of elimination of imaginaries by choosing our theory T to be a first-order theory coding finite sets and working in a large enough saturated model M of T.

October 23 - John Spoerl

Title: The Computational Content of Forcing

Abstract: Given information about a ground model, how much can you know about it’s forcing extensions? Given a forcing extension, how much can you know about its ground model? This talk will be a primer on forcing and a review of the main results of Hamkins, Miller and Williams’ 2020 paper: “Forcing as a Computational Process”.

October 30 - Chiara Travesset

Title: Mission-time LTL (MLTL) Formula Validation Via Regular Expressions

Abstract: Mission-time Linear Temporal Logic (mLTL) is an extension of propositional logic that includes temporal operators over finite intervals of time. A computation is a (finite) sequence consisting of a truth assignment for each propositional variable at each time step. This presentation will describe how we can use regular expressions to describe the structure of the computations that make a given mLTL formula true. We prove soundness and completeness, and also give an implemented algorithm (the WEST program) and analyze its complexity both theoretically and experimentally. We generate a test suite using control flow diagrams to robustly test the code. Finally, we present the REST theorem, which significantly simplifies certain sets of computations. This talk only requires basic familiarity with propositional logic and computer science.

November 6 - Antonio Nakid Cordero

Title: Martin's Conjecture in the Turing degrees and the Enumeration Degrees

Abstract: Martin's Conjecture is an attempt to formalize the empirical phenomenon that "naturally occurring" Turing degrees are well-ordered. In this talk, I'll go over how one can make that into a precise mathematical statement, a uniform version of the conjecture that has been fully resolved, and some of the complications of trying to propose a similar conjecture in the enumeration degrees. If time allows, I'll report on my ongoing project in this direction.

November 13 - Alice Vidrine

Title: Enumeration Weihrauch degrees and closed choice problems (Slides and Recording Passcode: C89k#5&0)

Abstract: In the Weihrauch degrees, we think of elements of Baire space as names for elements of mathematical objects. In the enumeration Weihrauch (eW) degrees we follow a similar idea, but our names may only carry positive information. It turns out that this coding by positive information separates several choice problems that are equivalent in the Weihrauch degrees, and in this talk we look at some of the more striking examples and their proofs. We close with some discussion of various directions for further research.

November 20 - Logan Heath

Title: A Proof Theorist's Guide to First Aid: CUT-Elimination and Other Results of Classical Proof Theory (Slides)

Abstract: When are correct proofs also "bad" proofs, how can we avoid them, and should we even try? In this talk we examine some early results of classical proof theory regarding the consistency of the sequent calculus for FOL and PA. In both cases our approach will be to avoid proofs which are technically correct, but also defective in some sense.

November 27 - Thanksgiving Break

December 4 - Mei Rose Connor

Title: When ‘And’ is Too Much and ‘Or’ is Not Enough

Abstract: This work was motivated by the seven–valued predication system of the Jaina religion, which states that no statement exists as an absolute. This system, known as Saptabhangivada in Sanskrit, has 7 truth values that are the non-empty subsets of the set {asti, nāsti, avaktavyah}. These words translate to: it is, it is not, and it is unassertible, respectively. This talk will explore how these seven truth values relate to each other, the problems encountered when trying to formalise this system, and the author’s current resolution of said problems using lattice theory and introducing an eighth truth value.

December 11 - Ang Li

Title: TBA

Abstract: TBA


Previous Years

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

Retrieved from "https://wiki.math.wisc.edu/index.php?title=Graduate_Logic_Seminar&oldid=25640"