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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 B215.
* '''Where:''' Van Vleck B235
* '''Organizers:''' [https://www.math.wisc.edu/~omer/ Omer Mermelstein]
* '''Organizer:''' Mariya Soskova


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 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]-->


== Spring 2020 - Tentative schedule ==
== Spring 2025 ==


=== January 28 - Talk by visitor - No seminar ===
The seminar will be run as a 1-credit seminar Math 975. In Spring 2025, we will finish last semester's topic on Higher Computability Theory.Once we are done students will present a logic topic of their choice (it could be original work, but does not have to be).  If you are not enrolled but would like to audit it, please contact [mailto:soskova@wisc.edu Mariya Soskova].
=== February 3 - Talk by visitor - No seminar ===
=== February 10 - No seminar (speaker was sick) ===


=== February 17 - James Hanson ===
Presentation Schedule: [https://docs.google.com/spreadsheets/d/1uRSaI1edJ5sepz57NV07ohIfBSKL9FgkvJvMAewk1ms/edit?usp=sharing Sign up here.]


Title: The Topology of Definable Sets in Continuous Logic
Notes on Higher Computability Theory: [https://uwmadison.box.com/s/j3xftdj1i70d4lblxhzswhg9e25ajcpq Download the notes here.] You will need your UW-login. Please, do not distribute these notes without permission from the author.


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


=== February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott ===
=== '''January 27 - Organizational Meeting and Sapir Ben-Shahar''' ===


'''Tejas Bhojraj''' - Quantum Kolmogorov Complexity.
Mariya Soskova will call for volunteers to sign up for presentations.  


Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.
Sapir Ben-Shahar will wrap up Section 5.1


'''Josiah Jacobsen-Grocott''' - A Characterization of Strongly $\eta$-Representable Degrees.
=== '''February 3 - Taeyoung Em''' ===


Abstract:
Taeyoung Em will present Section 5.3.  
$\eta$-representations are a way of coding sets in computable linear orders that were first
introduced by Fellner in his PhD thesis. Limitwise monotonic functions have been used to
characterize the sets with $\eta$-representations as well as the sets with subclasses of
$\eta$-representations except for the case of sets with strong $\eta$-representations, the only
class where the order type of the representation is unique.


We introduce the notion of a connected approximation of a set, a variation on $\Sigma^0_2$
=== '''February 10 - Hongyu Zhu''' ===
approximations. We use connected approximations to
give a characterization of the degrees with strong $\eta$-representations as well new
characterizations of the subclasses of $\eta$-representations with known characterizations.


=== March 2 - Patrick Nicodemus ===
Hongyu Zhu will present Section 5.3


Title: A Sheaf-theoretic generalization of Los's theorem
=== '''February 17 - Karthik Ravishankar''' ===


Abstract: Sheaf theory deals in part with the behavior of functions on a small open neighborhood of a point. As one chooses smaller and smaller open neighborhoods around a point, one gets closer to the limit - the "germ" of the function of the point. The relationship between the "finite approximation" (the function's behavior on a small, but not infinitesimal, neighborhood) and the "limit" (its infinitesimal behavior) is akin to the concept of reasoning with finite approximations that underlies forcing. Indeed, there is a natural forcing language that arises in sheaf theory - this is somewhat unsurprising as at a purely formal level, a sheaf is almost identical as a data structure to a Kripke model. We will demonstrate the applicability of this forcing language by giving a Los's theorem for sheaves of models.
'''Title:''' Strong minimal covers and the cupping property


=== March 9 - Noah Schweber ===
'''Abstract:''' A longstanding question in degree theory has been whether every minimal Turing degree has a strong minimal cover. Meanwhile a strong example of degrees without SMC's are those which have the cupping property. It is known that PA degrees have the cupping property, as do degrees with a certain amount of escaping power. On the other hand, it is known that being weak in the sense of being non DNC and Hyperimmune-free lets you have a SMC. Degrees with the cupping property are closed upwards while it is not known if degrees with SMC are closed downwards.  It is also not known if every degree either has the cupping property or a SMC. In this talk we will review several of these results and present techniques used to build SMCs.


=== March 16 - Spring break - No seminar ===
=== '''February 24 - Hongyu Zhu''' ===
=== March 23 - Two short talks - Harry Main-Luu and Daniel Belin ===
=== March 30 - TBD ===


=== April 6 - TBD ===
'''Title:''' Seeing the forest does not account for the trees


=== April 13 - Faculty at conference - No seminar ===
'''Abstract:''' Say a first-order theory (or a type) has bounded axiomatization if it has an axiomatization by <math>\forall_n</math>-formulas for some finite n. In this talk, we will discuss basic properties of theories and types with (or without) bounded axiomatizations, and in particular whether boundedness of theories implies that of types. (The meaning of the title will be explained in due time.)
=== April 20 - Harry Main-Luu ===
=== April 27 - Harry Main-Luu ===


=== '''March 3 -  Uri Andrews''' ===


'''Title:''' On the spectra of computable models of disintegrated strongly minimal theories with bounded ranks


== Fall 2019 ==
'''Abstract:'''  The spectrum of a strongly minimal theory characterizes which of its countable models have computable copies (indexed by their dimensions). We will focus on the disintegrated strongly minimal theories, i.e., where the algebraic closure of a set is the union of the algebraic closures of the elements of the set.


=== September 5 - Organizational meeting ===
Somewhere in the late aughts, Alice Medvedev and I proved that if a theory is disintegrated strongly minimal and has a finite signature, then either all models are computable, no models are computable, or only the prime model is computable. Steffen Lempp and I tried to push this sort of analysis past finite signatures and we have results about theories which are disintegrated strongly minimal and every symbol in the (infinite) signature has rank less than or equal to 1 in the theory (i.e., you cannot have R(a,b,\bar{z}) if a and b are algebraically independent). Over this past winter break, I found a strategy to bring (some of) this analysis to strongly minimal theories in infinite languages as long as there is some finite N so that every symbol has rank less than or equal to N. I'll describe this strategy, and depending on time, I might even present something that loosely resembles a proof.


=== September 9 - No seminar ===
=== '''March 10 - Logan Heath''' ===


=== September 16 - Daniel Belin ===
'''Title:''' Degree Spectra of Theories
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.
'''Abstract:''' I will discuss the notion of the degree spectrum of a theory, introduce a class of questions one might ask about such a thing, point to a few of the answers to such questions, and look a little more closely at one such spectrum to highlight the sorts of techniques that arise in the area.


=== September 23 - Daniel Belin ===


Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued
=== '''March 17 - Yiqing Wang''' ===


=== September 30 - Josiah Jacobsen-Grocott ===
'''Title:''' The compactness theorem is overrated


Title: Scott Rank of Computable Models
'''Abstract:''' Elementary classes, or first-order logic in general, are limited in their ability to capture many natural mathematical classes, such as locally finite groups and Archimedean ordered fields. Conversely, obtaining meaningful results in the generality of non-elementary classes can be impossible. In 1978, Shelah introduced the notion of Abstract Elementary Classes (AECs), providing a framework for studying classes that are not first-order axiomatizable yet still possess rich model-theoretic properties and carry the same 'test question'.


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.
In this talk, I will try to give an overview of AECs, prove Shelah’s Presentation Theorem, and highlight some open problems in this area.


=== October 7 - Josiah Jacobsen-Grocott ===
=== '''March 31 - Chiara Travesset''' ===


Title: Scott Rank of Computable Codels - Continued
'''Title:''' The Sacks Density Theorem


=== October 14 - Tejas Bhojraj ===
'''Abstract:'''  The Sacks Density Theorem states that between any two c.e. degrees, there are two incomparable c.e. degrees. I will present a detailed proof of this theorem.


Title: Solovay and Schnorr randomness for infinite sequences of qubits.
=== '''April 7th -  Taeyoung Em''' ===


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.
'''Title:''' Sets that encode themselves


=== October 23 - Tejas Bhojraj ===
'''Abstract:''' Introreducible sets were introduced by Dekker and Myhill. Mansfield proved that complementary retraceable sets are computable and Seetapun and Slaman proved that complementary introreducible sets are computable. In this talk, I will present some results of Appel and McLaughlin on regressive sets, and maybe some other results.


Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued
=== '''April 14th - Lucas Duckworth''' ===


Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.
'''Title:''' The Paris-Harrington Theorem and Computability Results for Ramsey Theory


=== October 28 - Two short talks ===
'''Abstract:''' The Paris-Harrington Theorem, a slight extension of the famous Finite Ramsey Theorem, was shown to be not provable in PA. This talk will explore this original proof, which proves this result using an interesting variety of arguments combining combinatorics and logic. If time permits, I will also speak about a few results of Jokusch, Specker, Yates, and others on various results using computability to prove properties about homogeneous sets from basic partitions in the original Finite Ramsey Theorem.


'''Iván Ongay Valverde''' - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)
=== '''April 21 -  Ang Li's defense''' ===


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):
'''Title:''' Computability-Theoretic Analysis of Ordered Groups, Logical Depth, and Introenumerability.  


- 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 $E$?
'''Abstract:''' The first part of this talk continues the study of connections between reverse mathematics and Weihrauch reducibility. In particular, we analyze the uniform computational power of problems formed from Maltsev’s classification of the order types of countable ordered groups. Several non-reducibility results are obtained via the first-order part—an interior algebraic operator on Weihrauch degrees that has attracted recent attention.
- 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.
In the second part, we turn to two measure and category questions: knowing that some classes of enumeration degrees have measure zero, what level of randomness can they have or must avoid as reasonable classes of randomness have measure one; knowing that the class of shallow sets is comeager, what level of genericity can deep sets have or must avoid as the classes of generic sets are comeager.


'''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.
=== '''April 28th - Sapir Ben-Shahar''' ===


=== November 4 - Two short talks ===
'''Title:''' Monadic definability and Matroids


'''Manlio Valenti''' - The complexity of closed Salem sets (20 minutes version)
'''Abstract:''' Matroids are combinatorial structures that generalise the idea of (in)dependence, such as linear independence of vectors in vector spaces over some field. Matroids arise in a number of different contexts, including from vectors in vector spaces, graphs, points in a geometry, as models of strongly minimal theories, in combinatorial optimization problems, in phylogenetic trees, and many more. This talk will focus on gain-graphic matroids, which are matroids that arise from group-labeled graphs. Gain-graphic matroids are important in the study of structural matroid theory. It turns out that for ``nice enough" classes of matroids, properties that are definable in monadic second-order logic can be recognized in polynomial time. Whether a class of gain-graphic matroids is definable or not depends on which group is chosen to label the graphs. I'll start with a brief introduction to matroids and monadic second order logic, and then describe recent progress on the definability question for gain-graphic matroids.


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.
== Fall 2024 ==


=== November 11 - Manlio Valenti ===
The seminar will be run as a 1-credit seminar Math 975 . In Fall 2024, the topic will be Higher Computability Theory. We will follow notes by Noam Greenberg. If you are not enrolled but would like to audit it, please contact [mailto:soskova@wisc.edu Mariya Soskova].


Title: The complexity of closed Salem sets (full length)
Presentation Schedule: [https://docs.google.com/spreadsheets/d/1ect-dgHdoHOgq4-5BGFiDh6pPThLfDg69Yg__-b_5RY/edit?usp=sharing Sign up here.]


Abstract:
Notes: [https://uwmadison.box.com/s/j3xftdj1i70d4lblxhzswhg9e25ajcpq Download the notes here.] You will need your UW-login. Please, do not distribute these notes without permission from the author.  
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 ===
<!--Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)-->


Title: A couple of summer results
=== '''September 9 - Organizational Meeting''' ===


Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.
Mariya Soskova will start with the first sections from the 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.
We will then assign speakers to dates and topics.


=== November 25 - Anniversary of the signing of the Treaty of Granada - No seminar ===
=== '''September 16 - Sections 1.2-1.4''' ===


=== December 2 - Anniversary of the Battle of Austerlitz - No seminar ===
Kanav Madhura will continue with Sections 1.2-1.4.


=== December 9 - Anniversary of the death of Pope Pius IV - No seminar  ===
=== '''September 23 - Sections 1.3-1.4 and 2.1-2.2''' ===


==Previous Years==
Kanav Madhura will continue with Sections 1.3-1.4. Lucas Duckworth will be ready with Sections 2.1 and 2.2 should there be time.
 
=== '''September 30 -  Sections 2.2 and 2.3-2.5''' ===
 
Lucas Duckworth will finish Section 2.2. Karthik Ravishankar will begin 2.3, 2.4, and 2.5.
=== '''October 7th -  Sections 2.4 and 2.5''' ===
 
Karthik Ravishankar will  finish, 2.4, and 2.5.  Liang Yu will give a talk at 4:00pm.
 
=== '''October 14th -  Sections 2.6 and 2.7''' ===
 
Bjarki Gunnarsson  will present Sections 2.6 and 2.7
 
=== '''October 21th -  Section 3.1''' ===
 
Karthik Ravishankar will present Section 3.1 
 
=== '''October 28th -  Sections 3.2 and 3.3''' ===
 
Karthik Ravishankar will finish Sections 3.2  and John Spoerl will begin Section 3.3
 
=== '''November 4th -  Sections 3.3 and 3.4''' ===
 
John Spoerl will finish Sections 3.3 and 3.4
 
=== '''November 11th -  Section 4.1''' ===
 
Antonion Nakid-Cordero will present Section 4.1
 
=== '''November 19th -  Sections 4.1 and 4.2''' ===
 
Start 4:00PM in VV901! Antonion Nakid-Cordero will continue with Section 4.1, Ang Li will begin Section 4.2.
 
 
=== '''November 25th -  Sections 4.2 and 4.3''' ===
 
Back to the usual time and place. Ang Li will begin Section 4.2.
 
=== '''December 2nd -  Section 4.3''' ===
 
Ang Li will present Section 4.3.
 
=== '''December 9nd -  Section 5.1''' ===
 
Last seminar for this semester. Sapir Ben-Shahar will begin Section 5.1
 
<!-- Template
 
=== '''September 18 - xxx''' ===
'''Title:''' TBA ([https://wiki.math.wisc.edu/images/***.pdf Slides])
 
'''Abstract:''' TBA
 
-->
 
== 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 18:10, 25 April 2025

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 3:30-4:30 PM
  • Where: Van Vleck B235
  • Organizer: Mariya Soskova

The talk schedule is arranged at the beginning of each semester. If you would like to participate, please contact the organizers.


Spring 2025

The seminar will be run as a 1-credit seminar Math 975. In Spring 2025, we will finish last semester's topic on Higher Computability Theory.Once we are done students will present a logic topic of their choice (it could be original work, but does not have to be). If you are not enrolled but would like to audit it, please contact Mariya Soskova.

Presentation Schedule: Sign up here.

Notes on Higher Computability Theory: Download the notes here. You will need your UW-login. Please, do not distribute these notes without permission from the author.


January 27 - Organizational Meeting and Sapir Ben-Shahar

Mariya Soskova will call for volunteers to sign up for presentations.

Sapir Ben-Shahar will wrap up Section 5.1

February 3 - Taeyoung Em

Taeyoung Em will present Section 5.3.

February 10 - Hongyu Zhu

Hongyu Zhu will present Section 5.3

February 17 - Karthik Ravishankar

Title: Strong minimal covers and the cupping property

Abstract: A longstanding question in degree theory has been whether every minimal Turing degree has a strong minimal cover. Meanwhile a strong example of degrees without SMC's are those which have the cupping property. It is known that PA degrees have the cupping property, as do degrees with a certain amount of escaping power. On the other hand, it is known that being weak in the sense of being non DNC and Hyperimmune-free lets you have a SMC. Degrees with the cupping property are closed upwards while it is not known if degrees with SMC are closed downwards. It is also not known if every degree either has the cupping property or a SMC. In this talk we will review several of these results and present techniques used to build SMCs.

February 24 - Hongyu Zhu

Title: Seeing the forest does not account for the trees

Abstract: Say a first-order theory (or a type) has bounded axiomatization if it has an axiomatization by [math]\displaystyle{ \forall_n }[/math]-formulas for some finite n. In this talk, we will discuss basic properties of theories and types with (or without) bounded axiomatizations, and in particular whether boundedness of theories implies that of types. (The meaning of the title will be explained in due time.)

March 3 - Uri Andrews

Title: On the spectra of computable models of disintegrated strongly minimal theories with bounded ranks

Abstract: The spectrum of a strongly minimal theory characterizes which of its countable models have computable copies (indexed by their dimensions). We will focus on the disintegrated strongly minimal theories, i.e., where the algebraic closure of a set is the union of the algebraic closures of the elements of the set.

Somewhere in the late aughts, Alice Medvedev and I proved that if a theory is disintegrated strongly minimal and has a finite signature, then either all models are computable, no models are computable, or only the prime model is computable. Steffen Lempp and I tried to push this sort of analysis past finite signatures and we have results about theories which are disintegrated strongly minimal and every symbol in the (infinite) signature has rank less than or equal to 1 in the theory (i.e., you cannot have R(a,b,\bar{z}) if a and b are algebraically independent). Over this past winter break, I found a strategy to bring (some of) this analysis to strongly minimal theories in infinite languages as long as there is some finite N so that every symbol has rank less than or equal to N. I'll describe this strategy, and depending on time, I might even present something that loosely resembles a proof.

March 10 - Logan Heath

Title: Degree Spectra of Theories

Abstract: I will discuss the notion of the degree spectrum of a theory, introduce a class of questions one might ask about such a thing, point to a few of the answers to such questions, and look a little more closely at one such spectrum to highlight the sorts of techniques that arise in the area.


March 17 - Yiqing Wang

Title: The compactness theorem is overrated

Abstract: Elementary classes, or first-order logic in general, are limited in their ability to capture many natural mathematical classes, such as locally finite groups and Archimedean ordered fields. Conversely, obtaining meaningful results in the generality of non-elementary classes can be impossible. In 1978, Shelah introduced the notion of Abstract Elementary Classes (AECs), providing a framework for studying classes that are not first-order axiomatizable yet still possess rich model-theoretic properties and carry the same 'test question'.

In this talk, I will try to give an overview of AECs, prove Shelah’s Presentation Theorem, and highlight some open problems in this area.

March 31 - Chiara Travesset

Title: The Sacks Density Theorem

Abstract: The Sacks Density Theorem states that between any two c.e. degrees, there are two incomparable c.e. degrees. I will present a detailed proof of this theorem.

April 7th - Taeyoung Em

Title: Sets that encode themselves

Abstract: Introreducible sets were introduced by Dekker and Myhill. Mansfield proved that complementary retraceable sets are computable and Seetapun and Slaman proved that complementary introreducible sets are computable. In this talk, I will present some results of Appel and McLaughlin on regressive sets, and maybe some other results.

April 14th - Lucas Duckworth

Title: The Paris-Harrington Theorem and Computability Results for Ramsey Theory

Abstract: The Paris-Harrington Theorem, a slight extension of the famous Finite Ramsey Theorem, was shown to be not provable in PA. This talk will explore this original proof, which proves this result using an interesting variety of arguments combining combinatorics and logic. If time permits, I will also speak about a few results of Jokusch, Specker, Yates, and others on various results using computability to prove properties about homogeneous sets from basic partitions in the original Finite Ramsey Theorem.

April 21 - Ang Li's defense

Title: Computability-Theoretic Analysis of Ordered Groups, Logical Depth, and Introenumerability.


Abstract: The first part of this talk continues the study of connections between reverse mathematics and Weihrauch reducibility. In particular, we analyze the uniform computational power of problems formed from Maltsev’s classification of the order types of countable ordered groups. Several non-reducibility results are obtained via the first-order part—an interior algebraic operator on Weihrauch degrees that has attracted recent attention.

In the second part, we turn to two measure and category questions: knowing that some classes of enumeration degrees have measure zero, what level of randomness can they have or must avoid as reasonable classes of randomness have measure one; knowing that the class of shallow sets is comeager, what level of genericity can deep sets have or must avoid as the classes of generic sets are comeager.


April 28th - Sapir Ben-Shahar

Title: Monadic definability and Matroids

Abstract: Matroids are combinatorial structures that generalise the idea of (in)dependence, such as linear independence of vectors in vector spaces over some field. Matroids arise in a number of different contexts, including from vectors in vector spaces, graphs, points in a geometry, as models of strongly minimal theories, in combinatorial optimization problems, in phylogenetic trees, and many more. This talk will focus on gain-graphic matroids, which are matroids that arise from group-labeled graphs. Gain-graphic matroids are important in the study of structural matroid theory. It turns out that for ``nice enough" classes of matroids, properties that are definable in monadic second-order logic can be recognized in polynomial time. Whether a class of gain-graphic matroids is definable or not depends on which group is chosen to label the graphs. I'll start with a brief introduction to matroids and monadic second order logic, and then describe recent progress on the definability question for gain-graphic matroids.


Fall 2024

The seminar will be run as a 1-credit seminar Math 975 . In Fall 2024, the topic will be Higher Computability Theory. We will follow notes by Noam Greenberg. If you are not enrolled but would like to audit it, please contact Mariya Soskova.

Presentation Schedule: Sign up here.

Notes: Download the notes here. You will need your UW-login. Please, do not distribute these notes without permission from the author.


September 9 - Organizational Meeting

Mariya Soskova will start with the first sections from the notes.

We will then assign speakers to dates and topics.

September 16 - Sections 1.2-1.4

Kanav Madhura will continue with Sections 1.2-1.4.

September 23 - Sections 1.3-1.4 and 2.1-2.2

Kanav Madhura will continue with Sections 1.3-1.4. Lucas Duckworth will be ready with Sections 2.1 and 2.2 should there be time.

September 30 - Sections 2.2 and 2.3-2.5

Lucas Duckworth will finish Section 2.2. Karthik Ravishankar will begin 2.3, 2.4, and 2.5.

October 7th - Sections 2.4 and 2.5

Karthik Ravishankar will finish, 2.4, and 2.5. Liang Yu will give a talk at 4:00pm.

October 14th - Sections 2.6 and 2.7

Bjarki Gunnarsson will present Sections 2.6 and 2.7

October 21th - Section 3.1

Karthik Ravishankar will present Section 3.1

October 28th - Sections 3.2 and 3.3

Karthik Ravishankar will finish Sections 3.2 and John Spoerl will begin Section 3.3

November 4th - Sections 3.3 and 3.4

John Spoerl will finish Sections 3.3 and 3.4

November 11th - Section 4.1

Antonion Nakid-Cordero will present Section 4.1

November 19th - Sections 4.1 and 4.2

Start 4:00PM in VV901! Antonion Nakid-Cordero will continue with Section 4.1, Ang Li will begin Section 4.2.


November 25th - Sections 4.2 and 4.3

Back to the usual time and place. Ang Li will begin Section 4.2.

December 2nd - Section 4.3

Ang Li will present Section 4.3.

December 9nd - Section 5.1

Last seminar for this semester. Sapir Ben-Shahar will begin Section 5.1


Previous Years

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

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