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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 an space focus principally in  practicing presentation skills or learning materials that are not usually presented on 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, 4:00 PM – 5:00 PM (unless otherwise announced).
* '''When:''' Mondays 3:30-4:30 PM
* '''Where:''' Van Vleck B235 (unless otherwise announced).
* '''Where:''' Van Vleck B235
* '''Organizers:''' [https://www.math.wisc.edu/~msoskova/ Mariya Soskava]
* '''Organizer:''' Mariya Soskova


Talks schedule are arrange and decide 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.


== Spring 2018 ==
<!--Sign up for the graduate logic seminar mailing list:  [mailto:join-grad-logic-sem@lists.wisc.edu join-grad-logic-sem@lists.wisc.edu]-->


=== January 29, Organizational meeting ===
== Spring 2025 ==


This day we decided the schedule for the semester.
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 5, [http://www.math.wisc.edu/~andrews/ Uri Andrews] ===
Presentation Schedule: [https://docs.google.com/spreadsheets/d/1uRSaI1edJ5sepz57NV07ohIfBSKL9FgkvJvMAewk1ms/edit?usp=sharing Sign up here.]


Title: Building Models of Strongly Minimal Theories - Part 1
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: Since I'm talking in the Tuesday seminar as well, I'll use the Monday seminar talk to do some background on the topic and some
<!--Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)-->
lemmas that will go into the proofs in Tuesday's talk. There will be (I hope) some theorems of interest to see on both days, and both on
the general topic of answering the following question: What do you need to know about a strongly minimal theory in order to compute
copies of all of its countable models. I'll start with a definition for strongly minimal theories and build up from there.


=== February 12, James Hanson ===
=== '''January 27 - Organizational Meeting and Sapir Ben-Shahar''' ===


Title: Finding Definable Sets in Continuous Logic
Mariya Soskova will call for volunteers to sign up for presentations.


Abstract: In order to be useful the notion of a 'definable set' in continuous logic is stricter than a naive comparison to discrete logic
Sapir Ben-Shahar will wrap up Section 5.1
would suggest. As a consequence, even in relatively tame theories there can be very few definable sets. For example, there is a
superstable theory with no non-trivial definable sets. As we'll see, however, there are many definable sets in omega-stable,
omega-categorical, and other small theories.


=== February 19, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] ===
=== '''February 3 -  Taeyoung Em''' ===


Title: Proper forcing
Taeyoung Em will present Section 5.3.


Abstract: Although a given forcing notion may have nice properties on its own, those properties might vanish when we apply it repeatedly.
=== '''February 10 -  Hongyu Zhu''' ===
Early preservation results (that is, theorems saying that the iteration of forcings with a nice property retains that nice property)
were fairly limited, and things really got off the ground with Shelah's invention of "proper forcing." Roughly speaking, a forcing is
proper if it can be approximated by elementary submodels of the universe in a particularly nice way. I'll define proper forcing and
sketch some applications.


=== February 26, Patrick Nicodemus ===
Hongyu Zhu will present Section 5.3


Title: A survey of computable and constructive mathematics in economic history
=== '''February 17 -  Karthik Ravishankar''' ===


=== March 5, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] ===
'''Title:''' Strong minimal covers and the cupping property


Title: Convexly Orderable Groups
'''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 12, [https://math.nd.edu/people/visiting-faculty/daniel-turetsky/ Dan Turetsky] (University of Notre Dame) ===
=== '''February 24 - Hongyu Zhu''' ===


Title: Structural Jump
'''Title:''' Seeing the forest does not account for the trees


=== March 19, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] ===
'''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.)


Title: Networks and degrees of points in non-second countable spaces
=== '''March 3 - Uri Andrews''' ===


=== April 2, Wil Cocke ===
'''Title:''' On the spectra of computable models of disintegrated strongly minimal theories with bounded ranks


Title: Characterizing Finite Nilpotent Groups via Word Maps
'''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.


Abstract: In this talk, we will examine a novel characterization of finite nilpotent groups using the probability distributions induced by word maps. In particular we show that a finite group is nilpotent if and only if every surjective word map has fibers of uniform size.
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.


=== April 9, Tejas Bhojraj ===
=== '''March 10 -  Logan Heath''' ===


Title: Quantum Randomness
'''Title:''' Degree Spectra of Theories


Abstract: I will read the paper by Nies and Scholz where they define a notion of algorithmic randomness for infinite sequences of quantum bits (qubits). This talk will cover the basic notions of quantum randomness on which my talk on Tuesday will be based.  
'''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.


=== April 16, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde] ===


Title: What can we say about sets made by the union of Turing equivalence classes?
=== '''March 17 -  Yiqing Wang''' ===


Abstract: It is well known that given a real number x (in the real line) the set of all reals that have the same Turing degree (we will call this a Turing equivalence class) have order type 'the rationals' and that, unless x is computable, the set is not a subfield of the reals. Nevertheless, what can we say about the order type or the algebraic structure of a set made by the uncountable union of Turing equivalence classes?
'''Title:''' The compactness theorem is overrated


This topic hasn't been deeply studied. In this talk I will focus principally on famous order types and answer whether they can be achieved or not. Furthermore, I will explain some possible connections with the automorphism problem of the Turing degrees.
'''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'.  


This is a work in progress, so this talk will have multiple open questions and opportunities for feedback and public participation.(hopefully).
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.


=== April 23, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] (Thesis Defense) Start 3:35 Room B231===
=== '''March 31 -  Chiara Travesset''' ===


Title: Cototal enumeration degrees and their applications to effective mathematics
'''Title:''' The Sacks Density Theorem


Abstract: The enumeration degrees measure the relative computational difficulty of enumerating sets of natural numbers. Unlike the Turing degrees, the enumeration degrees of a set and its complement need not be comparable. A set is total if it is enumeration above its complement. Taken together, the enumeration degrees of total sets form an embedded copy of the Turing degrees within the enumeration degrees. A set of natural numbers is cototal if it is enumeration reducible to its complement. Surprisingly, the degrees of cototal sets, the cototal degrees, form an intermediate structure strictly between the total degrees and the enumeration degrees.  
'''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.


Jeandel observed that cototal sets appear in a wide class of structures: as the word problems of simple groups, as the languages of minimal subshifts, and more generally as the maximal points of any c.e. quasivariety. In the case of minimal subshifts, the enumeration degree of the subshift's language determines the subshift's Turing degree spectrum: the collection of Turing degrees obtained by the points of the subshift. We prove that cototality precisely characterizes the Turing degree spectra of minimal subshifts: the degree spectra of nontrivial minimal subshifts are precisely the cototal enumeration cones. On the way to this result, we will give several other characterizations of the cototal degrees, including as the degrees of maximal anti-chain complements on <math>\omega^{<\omega}</math>, and as the degrees of enumeration-pointed trees on <math>2^{<\omega}</math>, and we will remark on some additional applications of these characterizations.
=== '''April 7th - Taeyoung Em''' ===


=== April 30, [http://www.math.uconn.edu/~westrick/ Linda Brown Westrick] (from University Of Connecticut) ===
'''Title:''' Sets that encode themselves


Title: TBA
'''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.


Abstract: TBA
=== '''April 14th - Lucas Duckworth''' ===


== Fall 2017 ==
'''Title:''' The Paris-Harrington Theorem and Computability Results for Ramsey Theory


=== September 11, Organizational meeting ===
'''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.


This day we decided the schedule for the semester.
=== '''April 21 -  Ang Li's defense''' ===


=== September 18, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] ===
'''Title:''' Computability-Theoretic Analysis of Ordered Groups, Logical Depth, and Introenumerability.  


Title: The Kunen inconsistency
'''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.


Abstract: While early large cardinal axioms were usually defined combinatorially - e.g., cardinals satisfying a version of Ramsey's
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.
theorem - later focus shifted to model-theoretic definitions, specifically definitions in terms of elementary embeddings of the
whole universe of sets. At the lowest level, a measurable cardinal is one which is the least cardinal moved (= critical point) by a
nontrivial elementary embedding from V into some inner model M.


There are several variations on this theme yielding stronger and stronger large cardinal notions; one of the most important is the
inclusion of *correctness properties* of the target model M. The strongest such correctness property is total correctness: M=V. The
critical point of an elementary embedding from V to V is called a *Reinhardt cardinal*. Shortly after their introduction in Reinhardt's
thesis, however, the existence of a Reinhardt cardinal was shown to be inconsistent with ZFC.


I'll present this argument, and talk a bit about the role of choice.
=== '''April 28th -  Sapir Ben-Shahar''' ===


=== September 25, [https://sites.google.com/a/wisc.edu/schweber/ Noah Schweber] ===
'''Title:''' Monadic definability and Matroids


Title: Hindman's theorem via ultrafilters
'''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.


Abstract: Hindman's theorem is a Ramsey-type theorem in additive combinatorics: if we color the natural numbers with two colors, there is an infinite set such that any *finite sum* from that set has the same color as any other finite sum. There are (to my knowledge) two proofs of Hindman's theorem: one of them is a complicated mess of combinatorics, and the other consists of cheating wildly. We'll do.


=== October 2, James Hanson ===


Title: The Gromov-Hausdorff metric on type space in continuous logic
== Fall 2024 ==


Abstract: The Gromov-Hausdorff metric is a notion of the 'distance' between two metric spaces. Although it is typically studied in the context of compact or locally compact metric spaces, the definition is sensible even when applied to non-compact metric spaces, but in that context it is only a pseudo-metric: there are non-isomorphic metric spaces with Gromov-Hausdorff distance 0. This gives rise to an equivalence relation that is slightly coarser than isomorphism. There are continuous first-order theories which are categorical with regards to this equivalence relation while failing to be isometrically categorical, so it is natural to look for analogs of the Ryll-Nardzewski theorem and Morley's theorem, but before we can do any of that, it'll be necessary to learn about the "topometric" structure induced on type space by the Gromov-Hausdorff metric.
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].


=== October 9, James Hanson ===
Presentation Schedule: [https://docs.google.com/spreadsheets/d/1ect-dgHdoHOgq4-5BGFiDh6pPThLfDg69Yg__-b_5RY/edit?usp=sharing Sign up here.]


Title: Morley rank and stability in continuous logic
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.


Abstract: There are various ways of counting the 'size' of subsets of metric spaces. Using these we can do a kind of Cantor-Bendixson analysis on type spaces in continuous first-order theories, and thereby define a notion of Morley rank. More directly we can define
<!--Zoom link for remote attendance: https://uwmadison.zoom.us/j/96168027763?pwd=bGdvL3lpOGl6QndQcG5RTFUzY3JXQT09 (Meeting ID: 961 6802 7763, Password: 975f23)-->
> the 'correct' notion of stability in the continuous setting. There are also natural Gromov-Hausdorff (GH) analogs of these notions. With this we'll prove that inseparably categorical theories have atomic models over arbitrary sets, which is an important step in the proof of Morley's theorem in this setting. The same proof with essentially cosmetic changes gives that inseparably GH-categorical theories have 'GH-atomic' models over arbitrary sets, but GH-atomic models fail to be GH-unique in general.


=== October 23, [http://www.math.wisc.edu/~makuluni/ Tamvana Makulumi] ===
=== '''September 9 - Organizational Meeting''' ===


Title: Boxy sets in ordered convexly-orderable structures.
Mariya Soskova will start with the first sections from the notes.


=== October 30, [http://www.math.wisc.edu/~ongay/ Iván Ongay-Valverde] ===
We will then assign speakers to dates and topics.


Title: Dancing SCCA and other Coloring Axioms
=== '''September 16 -  Sections 1.2-1.4''' ===


Abstract: In this talk I will talk about some axioms that are closely related to SOCA (Semi Open Coloring Axiom), being the main protagonist SCCA (Semi Clopen Coloring Axiom). I will give a motivation on the statements of both axioms, a little historic perspective and showing that both axioms coincide for separable Baire spaces. This is a work in progress, so I will share some open questions that I'm happy to discuss.
Kanav Madhura will continue with Sections 1.2-1.4.  


=== November 6, Wil Cocke ===
=== '''September 23 -  Sections 1.3-1.4 and 2.1-2.2''' ===


Title: Two new characterizations of nilpotent groups
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.


Abstract: We will give two new characterizations of finite nilpotent groups. One using information about the order of products of elements of prime order and the other using the induced probability distribution from word maps.
=== '''September 30 -  Sections 2.2 and 2.3-2.5''' ===


Or...
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''' ===


Title: Centralizing Propagating Properties of Groups
Karthik Ravishankar will  finish, 2.4, and 2.5.  Liang Yu will give a talk at 4:00pm.


Abstract: We will examine some sentences known to have finite spectrum when conjoined with the theory of groups. Hopefully we will be able to find new examples.  
=== '''October 14th -  Sections 2.6 and 2.7''' ===


=== November 13, [https://www.math.wisc.edu/~lempp/ Steffen Lempp] ===
Bjarki Gunnarsson  will present Sections 2.6 and 2.7


Title: The computational complexity of properties of finitely presented groups
=== '''October 21th -  Section 3.1''' ===


Abstract: I will survey index set complexity results on finitely presented groups.
Karthik Ravishankar will present Section 3.1 


=== November 20, [http://www.math.wisc.edu/~mccarthy/ Ethan McCarthy] ===
=== '''October 28th -  Sections 3.2 and 3.3''' ===


Title: Strong Difference Randomness
Karthik Ravishankar will finish Sections 3.2  and John Spoerl will begin Section 3.3


Abstract: The difference randoms were introduced by Franklin and Ng to characterize the incomplete Martin-Löf randoms. More recently, Bienvenu and Porter introduced the strong difference randoms, obtained by imposing the Solovay condition over the class of difference tests. I will give a Demuth test characterization of the strong difference randoms, along with a lowness characterization of them among the Martin-Löf randoms.  
=== '''November 4th - Sections 3.3 and 3.4''' ===


=== December 4, Tejas Bhojraj ===
John Spoerl will finish Sections 3.3 and 3.4  


Title: Quantum Algorithmic Randomness
=== '''November 11th -  Section 4.1''' ===


Abstract: I will discuss the recent paper by Nies and Scholz where they define quantum Martin-Lof randomness (q-MLR) for infinite sequences of qubits. If time permits, I will introduce the notion of quantum Solovay randomness and show that it is equivalent to q-MLR in some special cases.
Antonion Nakid-Cordero will present Section 4.1


=== December 11, Grigory Terlov ===
=== '''November 19th -  Sections 4.1 and 4.2''' ===


Title: The Logic of Erdős–Rényi Graphs
Start 4:00PM in VV901! Antonion Nakid-Cordero will continue with Section 4.1, Ang Li will begin Section 4.2.


==Previous Years==


The schedule of talks from past semesters can be found [[Logic Graduate Seminar, previous semesters|here]].
=== '''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]].

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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