Graduate Logic Seminar, previous semesters

Spring 2024

The seminar will be run as a 1-credit seminar Math 975 . In Spring 2024, the topic will be forcing constructions in computability theory. If you are not enrolled but would like to audit it, please contact Steffen Lempp and Hongyu Zhu.

Presentation Schedule: https://docs.google.com/spreadsheets/d/1JC6glG_soNLtaMQWaAuADlUu8dh2eJ0NL-MaUr7-nOk/edit?usp=sharing

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

January 29 - Organizational Meeting[edit | edit source]

Steffen Lempp will give an overview and present some very basic forcing construction.

We will then assign speakers to dates and topics.

February 5 - Taeyoung Em[edit | edit source]

Title: Introduction to forcing

Abstract: We introduce new definitions and properties regarding forcing.

February 12 - Hongyu Zhu[edit | edit source]

Title: Slaman-Woodin Forcing and the Theory of Turing Degrees

Abstract: We will discuss how to use Slaman-Woodin forcing to interpret true second(first, resp.)-order arithmetic in the Turing degrees (Turing degrees below 0', resp.), thereby showing they have the same Turing degree.

February 19 - John Spoerl[edit | edit source]

Title: Forcing with Trees - Spector's and Sack's Minimal Degrees

Abstract: We'll take a look at Spector's forcing which uses perfect trees as conditions. Then we'll see where we might make some improvements which leads to Sack's sharpening of Spector's theorem: there is a minimal degree below 0'.

February 26 - Karthik Ravishankar[edit | edit source]

Title: The 3 element chain as an initial segment of the Turing Degrees

Abstract: In this talk, we'll look at the construction of a minimal degree with a strong minimal cover which shows that the three-element chain can be embedded as an initial segment of the Turing Degrees. The construction builds off ideas of Spector's minimal degree with stronger assumptions on the forcing conditions used. If time permits, we'll also talk about Copper's Jump Inversion building off Sack's construction.

March 4 - Karthik Ravishankar[edit | edit source]

Title: Bushy Tree forcing and constructing a minimal degree which is DNC

Abstract: We shall look at a forcing technique called Bushy Tree forcing using it to show that there is no uniform way to compute a DNC_2 from a DNC_3 function and that there is a DNC function that is weak in the sense that it does not compute a computably bounded DNC function. We present a few other results along these lines and sketch the construction of a minimal degree that is DNC relative to any given oracle using bushy tree forcing.

March 11 - Josiah Jacobsen-Grocott[edit | edit source]

Title: A uniformly e-pointed tree on Baire space without dead ends that is not of cototal degree

Abstract: A set is cototal if it is enumeration reducible to its complement. A tree is e-point if every path on the tree can enumerate the tree. McCathy proved that these notions are equivalent up to e-degree when considering e-pointed trees on cantor space. This fails when considering trees on Baire space. We give an example of a simple forcing construction that produces e-pointed trees on Baire space. We carefully analyze this forcing partial order to prove that generic e-pointed trees without dead ends are not of cototal degree.

March 18 - Alice Vidrine[edit | edit source]

Title: There is no non-computable bi-introreducible set

Abstract: A set is said to be bi-introreducible if it can be computed by any of its infinite subsets, or any infinite subset of its complement. This talk will detail a Matthias forcing construction used to prove a theorem by Seetapun which implies that the bi-introreducible sets are exactly the computable sets.

April 1 - Hongyu Zhu[edit | edit source]

Title: The Conservativeness of WKL_0 over RCA_0 for -formulas

Abstract: We will see how to use forcing to construct models of WKL_0 from models of RCA_0 while preserving certain arithmetical truths, thereby showing that WKL_0 is -conservative over RCA_0.

April 15 - Ang Li[edit | edit source]

Title: Steel Forcing without Generalized Ramified Forcing Language

Abstract: In this talk, we will introduce Steel forcing, also known as "forcing with tagged trees", using a restricted infinitary language. We will prove the retagging lemma and show the example that Well-Foundededness and Unique Branch are not Borel separable. If time permits, we will talk about another example in descriptive set theory that uses steel forcing purely topologically.

April 22 - Antonio Nakid Cordero[edit | edit source]

Title: Kumabe-Slaman Forcing and Definability of the Turing Jump

Abstract: Slaman and Woodin's Analysis of the automorphisms of the Turing Degrees gave as a consequence the definability of the double jump. We will prove a Posner-Robinson type theorem for n-CEA operators by Kumabe-Slaman forcing that, together with the definability of the double jump, gives the definability of the single jump.

April 29 - John Spoerl[edit | edit source]

Title: Forcing in Set Theory

Abstract: Despite the theme of this seminar being about computability-theoretic forcing, the forcing method is most famously used in set theory to prove various consistency results. The most important use of forcing was its invention by Paul Cohen to show the independence of the continuum hypothesis and the axiom of choice from ZFC. I'll discuss the subtleties and development of forcing in set theory and sketch the method by which one can build and control new models of ZF(C).

Fall 2023

The seminar was run as a 1-credit seminar Math 975 in Fall 2023.

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.

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 - NO MEETING

Note: The originally scheduled talk has been moved to Tuesday, December 12 (at the usual logic seminar time&location). See https://people.math.wisc.edu/logic/seminar.html for more details.

Spring 2023

The seminar was run as a 1-credit seminar Math 975 in spring, check out the Canvas page. The topic was priority arguments.

Fall 2022

September 12 - Organizational Meeting

We will meet to assign speakers to dates.

September 19 - Karthik Ravishankar

Title: Lowness for Isomorphism (Slides)

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.

September 26 - Antonio Nakid Cordero

Title: When Models became Polish: an introduction to the Topological Vaught Conjecture (Slides)

Abstract: Vaught's Conjecture, originally asked by Vaught in 1961, is one of the most (in)famous open problems in mathematical logic. The conjecture is that a complete theory on a countable language must either have countably-many or continuum-many non-isomorphic models. In this talk, we will discuss some of the main ideas that surround this conjecture, with special emphasis on a topological generalization in terms of the continuous actions of Polish groups.

October 10 - Yunting Zhang

Title: Some History of Logic (Slides)

Abstract: The lives of great thinkers are sometimes overshadowed by their achievements-and there is perhaps no better illustration of this phenomenon than the life and work of Gödel. Take a look at Gödel's own timeline and see how wars and other mathematicians influenced him.

October 17 - Alice Vidrine

Title: Local operators, bilayer Turing reducibility, and enumeration Weihrauch degrees (Slides)

Abstract: Realizability toposes have a rich variety of subtoposes, corresponding to their local operators. These local operators are somewhat difficult to study in their usual form, which seems far removed from the usual objects of computability theoretic study. Recent work by Takayuki Kihara has given a characterization of the local operators on the effective topos in computability theoretic terms related to Weihrauch reduction, and which generalizes to several other realizability toposes of possible interest to computability theorists. This narrative-focused talk outlines what a realizability topos looks like, what local operators are, what Kihara's bilayer Turing reduction looks like, and how this leads to preliminary questions about a relative of the Weihrauch degrees based on enumeration reduction.

October 24 - Hongyu Zhu

Title: Investigating Natural Theories through the Consistency Operator (Slides)

Abstract: The phenomenon that "natural" theories tend to be linearly ordered in terms of consistency strength is a long-standing mystery. One approach to solving the problem is Martin's Conjecture, which roughly claims that the only natural functions on the Turing degrees are transfinite iterates of the Turing jump. In this talk we will focus on a similar approach, working inside the Lindenbaum algebra of elementary arithmetic instead of the Turing degrees. Here, the consistency operator takes the role of the jump. We will see that while some nice analogous claims can be established, there are also counterexamples that prevent us from strengthening the results in various ways.

November 7 - John Spoerl

Title: Universal Algorithms

Abstract: Among the many ways we can flex Gödel’s Incompleteness theorems, there is one that feels especially strange: there is a partial computable function F, such that F gives no output in the standard model of arithmetic, but for any function G on natural numbers (computable or not) there is a non-standard model in which F behaves exactly as G. I’ll discuss some related arguments and some philosophical questions this may raise about our notions of finiteness and determinism.

November 14 - Josiah Jacobsen-Grocott

Title: The Paris-Harrington Theorem

Abstract: In this talk, I will present the proof of the Paris-Harrington theorem. The Paris-Harrington theorem states that over PA the consistency of Peano arithmetic is equivalent to a strengthening of the finite Ramsey theorem. This was the first example of a result from "ordinary mathematics" that can not be proven by PA. The aim of this talk is to cover the main logical steps in this proof and give some of the combinatorics.

November 21 - Karthik Ravishankar

Title: The computing power of Baire space vs Cantor Space

Abstract: Generic Muchnik reducibility extends the notion of Muchnik reducibility to the uncountable setting. In this talk we'll look at some recent work which comes up with another technique of constructing a structure of degree strictly between Cantor space and Baire Space. We'll see that there is a generic copy of Cantor space which computes no escaping function while every copy of Baire space computes a dominating function. We then construct a structure of intermediate degree which always computes an escaping function but no dominating function.

November 28 - Logan Heath

Title: A Mathematical Analysis of Theories of Generative Grammar: The Peters-Ritchie Theorem (Slides)

Abstract: This is a preview of a talk intended primarily for undergraduate students in linguistics who have just completed a first course in syntax. The goal of the talk is to spark interest in applications of mathematical techniques to the study of theories of syntax. We will focus on the Peters-Ritchie Theorem which shows that transformational grammars can generate languages which are c.e., but not computable.

December 12 - Yuxiao Fu

Title: Brief Introduction to Modal Logic (Slides)

Abstract: In this talk, I plan to give a mathematical introduction to modal logic and its applications, covering its syntax and semantics, Kripke frames and models, bisimulations, filtrations, frame definability, automatic first-order correspondence, canonical models, and conclude with the completeness of normal modal logics.

Spring 2022

The graduate logic seminar this semester was run as MATH 975. We covered the first 6 parts of Tin Lok Wong's notes.

February 1 - Steffen Lempp

I will give an overview of the topics we will cover:

1. the base theory PA^- and the induction and bounding axioms for Sigma_n-formulas, and how they relate to each other,

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

3. the Grzegorczyk hierarchy of fast-growing functions,

4. end extensions and cofinal extensions,

5. recursive saturation and resplendency,

6. standard systems and coded types,

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

8. Gaifman's theorem that every model of PA has a minimal elementary end extension.

I will sketch the basic definitions and state the main theorems, in a form that one can appreciate without too much background.

February 8 - Karthik Ravishankar

Title: Collection axioms

We will discuss parts 1 and 2 of Wong's notes.

February 15 - Karthik Ravishankar, Yunting Zhang

Title: Collection axioms/The Weak König Lemma

Karthik will finish part 2 of Wong's notes. Then Yunting will start on part 3 of Wong's notes.

February 22 - Yunting Zhang

Title: The Weak König Lemma

We will finish part 3 of Wong's notes.

March 22 - Ang Li

Title: The Arithmetized Completeness Theorem

We will discuss part 4 of Wong's notes.

March 29 - Ang Li

Title: The Arithmetized Completeness Theorem

We will finish part 4 of Wong's notes.

April 5 - Antonio Nakid Cordero

Title: Semiregular cuts

We will start on part 5 of Wong's notes.

April 12 - Antonio Nakid Cordero/Alice Vidrine

Title: Semiregular cuts/End and cofinal extensions

We will finish part 5 of Wong's notes and then start on part 6.

April 19 - Alice Vidrine

Title: End and cofinal extensions

We will finish part 6 of Wong's notes.

Fall 2021

To see what's happening in the Logic qual preparation sessions click here.

September 28 - Ouyang Xiating

Title: First-order logic, database and consistent query answering

Abstract: Databases are a crucial component of many (if not all) modern applications. In reality, the data stored are often dirty and contain duplicated/missing entries, and it is a natural practice to clean the data first before executing the query. However, the same query might return different answers on different cleaned versions of the dataset. It is then helpful to compute the consistent answers: the query answers that will always be returned, regardless of how the dirty data is cleaned. In this talk, we first introduce the connection between first-order logic and query languages on databases, and then discuss the problem of Consistent Query Answering (CQA): How to compute consistent answers on dirty data? Finally, we show when the CQA problem can be solved using first-order logic for path queries.

October 12 - Karthik Ravishankar

Title: Notions of randomness for subsets of the Natural Numbers

Abstract: There are a number of notions of randomness of sets of natural numbers. These notions have been defined based on what a 'random object' should behave like such as being 'incompressible' or being 'hard to predict' etc. There is often a interplay between computability and randomness aspects of subsets of natural numbers. In this talk we motivate and present a few different notions of randomness and compare their relative strength.

November 9 - Antonio Nakid Cordero

Title: Martin's Conjecture: On the uniqueness of the Turing jump

Abstract: The partial order of the Turing degrees is well-known to be extremely complicated. However, all the Turing degrees that appear "naturally" in mathematics turn out to be well-ordered. In the '70s, Martin made a sharp conjecture explaining this phenomenon, the prime suspect: the Turing jump. This talk will explore the precise statement of Martin's conjecture and the interesting mathematics that surround it.

November 23 - Antonio Nakid Cordero

Title: Two Perspectives on Martin's Conjecture.

Abstract: This time we will dive deeper into the recent developments around Martin's Conjecture. We will focus on two main themes: the uniformity assumption, and the interaction of Martin's conjecture with the theory of countable Borel equivalence relations.

December 7 - John Spoerl

Title: Cardinals Beyond Choice and Inner Model Theory

Abstract: This talk will be a general introduction and overview of large cardinal axioms which violate the axiom of choice and their impact on the project of inner model theory.

Spring 2021

February 16 3:30PM - Short talk by Sarah Reitzes (University of Chicago)

Title: Reduction games over $\mathrm{RCA}_0$

Abstract: In this talk, I will discuss joint work with Damir D. Dzhafarov and Denis R. Hirschfeldt. Our work centers on the characterization of problems P and Q such that P $\leq_{\omega}$ Q, as well as problems P and Q such that $\mathrm{RCA}_0 \vdash$ Q $\to$ P, in terms of winning strategies in certain games. These characterizations were originally introduced by Hirschfeldt and Jockusch. I will discuss extensions and generalizations of these characterizations, including a certain notion of compactness that allows us, for strategies satisfying particular conditions, to bound the number of moves it takes to win. This bound is independent of the instance of the problem P being considered. This allows us to develop the idea of Weihrauch and generalized Weihrauch reduction over some base theory. Here, we will focus on the base theory $\mathrm{RCA}_0$. In this talk, I will explore these notions of reduction among various principles, including bounding and induction principles.

March 23 4:15PM - Steffen Lempp

Title: Degree structures and their finite substructures

Abstract: Many problems in mathematics can be viewed as being coded by sets of natural numbers (as indices). One can then define the relative computability of sets of natural numbers in various ways, each leading to a precise notion of “degree” of a problem (or set). In each case, these degrees form partial orders, which can be studied as algebraic structures. The study of their finite substructures leads to a better understanding of the partial order as a whole.

March 30 4PM - Alice Vidrine

Title: Categorical logic for realizability, part I: Categories and the Yoneda Lemma

Abstract: An interesting strand of modern research on realizability--a semantics for non-classical logic based on a notion of computation--uses the language of toposes and Grothendieck fibrations to study mathematical universes whose internal notion of truth is similarly structured by computation. The purpose of this talk is to establish the basic notions of category theory required to understand the tools of categorical logic developed in the sequel, with the end goal of understanding the realizability toposes developed by Hyland, Johnstone, and Pitts. The talk will cover the definitions of category, functor, natural transformation, adjunctions, and limits/colimits, with a heavy emphasis on the ubiquitous notion of representability.

Link to slides

April 27 4PM - Alice Vidrine

Title: Categorical logic for realizability, part II

Abstract: Realizability is an approach to semantics for non-classical logic that interprets propositions by sets of abstract computational data. One modern approach to realizability makes heavy use of the notion of a topos, a type of category that behaves like a universe of non-standard sets. In preparation for introducing realizability toposes, the present talk will be a brisk introduction to the notion of a topos, with an emphasis on their logical aspects. In particular, we will look at the notion of a subobject classifier and the internal logic to which it gives rise.

Fall 2020

September 14 - Josiah Jacobsen-Grocott

Title: Degrees of points in topological spaces

Abstract: An overview of some results from Takayuki Kihara, Keng Meng Ng, and Arno Pauly in their paper Enumeration Degrees and Non-Metrizable Topology. We will look at a range of topological spaces and the corresponding classes in the enumeration degrees as well as ways in which we can distinguish the type of classes using the separation axioms.

September 28 - James Hanson

Title: The Semilattice of Definable Sets in Continuous Logic

Abstract: After an analysis-free exposition of definable sets in continuous logic, we will present a fun, illustrated proof that any finite bounded lattice can be the poset of definable subsets of $S_1(T)$ for a continuous theory $T$.

October 5 - Tejas Bhojraj from 3:30PM-4:00PM

Title: A Levin-Schnorr type result for Weak Solovay random states.

Abstract: We look at the initial-segment complexity of Weak Solovay quantum random states using MK, a prefix-free version of quantum Kolmogorov complexity. The statement of our result is similar to the Levin-Schnorr theorem in classical algorithmic randomness.

November 9 - Karthik Ravishankar

Title: Elementary submodels in infinite combinatorics

Abstract: The usage of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. In the first part of the talk, we quickly cover the basic concepts involved for proving results using elementary submodels, and move on to provide two examples of application of the technique to prove two popular results from set theory: The Delta System lemma and the Fodors Pressing down lemma . We provide both the classical proof as well as a proof using elementary submodels to contrast the two approaches.

November 16 - Karthik Ravishankar

Title: Elementary submodels in infinite combinatorics, part II

Abstract: In the second part of the talk, we give a proof Fodors Pressing down lemma, along with an overview of the slightly larger proof of the Nash Williams theorem which states that a graph is decomposable as a disjoint union of cycles if and only if it has no odd cut.

Tuesday, November 24 - Tonicha Crook (Swansea University) from 9:00AM-10:00AM

Title: The Weihrauch Degree of Finding Nash Equilibria in Multiplayer Games

Abstract: Is there an algorithm that takes a game in normal form as input, and outputs a Nash equilibrium? If the payoffs are integers, the answer is yes, and a lot of work has been done in its computational complexity. If the payoffs are permitted to be real numbers, the answer is no, for continuity reasons. It is worthwhile to investigate the precise degree of non-computability (the Weihrauch degree), since knowing the degree entails what other approaches are available (eg, is there a randomized algorithm with positive success change?). The two player case has already been fully classified, but the multiplayer case remains open and is addressed here. As well as some insight into finding the roots of polynomials, which is essential in our research. An in-depth introduction to Weihrauch Reducibility will be included in the presentation, along with a small introduction to Game Theory.

November 30 - Yvette Ren

Title: Enumeration Degrees and Topology

Abstract: I will introduce three classes of enumeration degrees: co-dcea degrees, telegraph-cototal degrees and cylinder-cototal degrees and discuss the relations between each other. Selected results from Kihara, Pauly, and Ng’s paper Enumeration Degrees and Non-Metrizable Topology and their alternative proofs will be presented.

Spring 2020

February 17 - James Hanson

Title: The Topology of Definable Sets in Continuous Logic

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.

February 24 - Two short talks - Tejas Bhojraj and Josiah Jacobsen-Grocott

Tejas Bhojraj - Quantum Kolmogorov Complexity.

Abstract: We define a notion of quantum Kolmogorov complexity and relate it to quantum Solovay and quantum Schnorr randomness.

Josiah Jacobsen-Grocott - A Characterization of Strongly $\eta$-Representable Degrees.

Abstract: $\eta$-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 $\Sigma^0_2$ 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

Title: A Sheaf-theoretic generalization of Los's theorem

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.

March 9 - Noah Schweber

Title: Algebraic logic and algebraizable logics

Abstract: Arguably the oldest theme in what we would recognize as "mathematical logic" is the algebraic interpretation of logic, the most famous example of this being the connection between (classical) propositional logic and Boolean algebras. But underlying the subject of algebraic logic is the implicit assumption that many logical systems are "satisfyingly" interpreted as algebraic structures. This naturally hints at a question, which to my knowledge went unasked for a surprisingly long time: when does a logic admit a "nice algebraic interpretation?"

Perhaps surprisingly, this is actually a question which can be made precise enough to treat with interesting results. I'll sketch what is probably the first serious result along these lines, due to Blok and Pigozzi, and then say a bit about where this aspect of algebraic logic has gone from there.

Fall 2019

September 16 - Daniel Belin

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.

September 23 - Daniel Belin

Title: Lattice Embeddings of the m-Degrees and Second Order Arithmetic - Continued

September 30 - Josiah Jacobsen-Grocott

Title: Scott Rank of Computable Models

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.

October 7 - Josiah Jacobsen-Grocott

Title: Scott Rank of Computable Codels - Continued

October 14 - Tejas Bhojraj

Title: Solovay and Schnorr randomness for infinite sequences of qubits.

Abstract : We define Solovay and Schnorr randomness in the quantum setting. We then prove quantum versions of the law of large numbers and of the Shannon McMillan Breiman theorem (only for the iid case) for quantum Schnorr randoms.

October 23 - Tejas Bhojraj

Title: Solovay and Schnorr randomness for infinite sequences of qubits - continued

Unusual time and place: Wednesday October 23, 4:30pm, Van Vleck B321.

October 28 - Two short talks - Iván Ongay Valverde and James Earnest Hanson

Iván Ongay Valverde - Exploring different versions of the Semi-Open Coloring Axiom (SOCA)

In 1985, Avraham, Rubin and Shelah published an article where they introduced different coloring axioms. The weakest of them, the Semi-Open Coloring Axiom (SOCA), states that given an uncountable second countable metric space, $E$, and $W\subseteq E^{\dagger}:=E\times E\setminus \{(x, x) :x \in E\}$ open and symmetric, there is an uncountable subset $H\subseteq E$ such that either $H^{\dagger}\subseteq W$ or $H^{\dagger}\cap W=\emptyset$. We say that $W$ is an open coloring and $H$ is a homogeneous subset of $E$. This statement contradicts CH but, as shown also by Avraham, Rubin and Shelah, it is compatible with the continuum taking any other size. This classic paper leaves some questions open (either in an implicit or an explicit way):

- Is the axiom weaker if we demand that $W$ is clopen? - If the continuum is bigger than $\aleph_2$, can we ask that $H$ has the same size as $E$? - Can we expand this axiom to spaces that are not second countable and metric?

These questions lead to different versions of SOCA. In this talk, we will analyze how they relate to the original axiom.

James Earnest Hanson - Strongly minimal sets in continuous logic

The precise structural understanding of uncountably categorical theories given by the proof of the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an $\omega$-stable theory. Finally, we will examine the extent to which we recover a Baldwin-Lachlan style characterization in the presence of strongly minimal sets.

November 4 - Two short talks - Manlio Valenti and Patrick Nicodemus

Manlio Valenti - The complexity of closed Salem sets (20 minutes version)

A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets.
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.

Patrick Nicodemus - Proof theory of Second Order Arithmetic and System F

A central theme in proof theory is to show that some formal system has the property that whenever A is provable, there is a proof of A in "normal form" - a direct proof without any detours. Such results have numerous and immediate consequences - often consistency follows as an easy corollary. The Curry Howard correspondence describes of equivalences between normalization of proofs and program termination in typed lambda calculi. We present an instance of this equivalence, between the proof theory of intuitionistic second order arithmetic and the second order polymorphic lambda calculus of Girard and Reynolds, aka system F.

November 11 - Manlio Valenti

Title: The complexity of closed Salem sets (full length)

Abstract: A central notion in geometric measure theory is the one of Hausdorff dimension. As a consequence of Frostman's lemma, the Hausdorff dimension of a Borel subset A of the Euclidean n-dimensional space can be determined by looking at the behaviour of probability measures with support in A. The possibility to apply methods from Fourier analysis to estimate the Hausdorff dimension gives birth to the notion of Fourier dimension. It is known that, for Borel sets, the Fourier dimension is less than or equal to the Hausdorff dimension. The sets for which the two notions agree are called Salem sets.
In this talk we will study the descriptive complexity of the family of closed Salem subsets of the real line.

November 18 - Iván Ongay Valverde

Title: A couple of summer results

Abstract: Lately, I have been studying how subsets of reals closed under Turing equivalence behave through the lenses of algebra, measure theory and orders.

In this talk I will classify which subsets of reals closed under Turing equivalence generate subfields or $\mathbb{Q}$-vector spaces of $\mathbb{R}$. We will show that there is a non-measurable set whose Turing closure becomes measurable (and one that stays non-measurable) and, if we have enough time, we will see a model where there are 5 possible order types for $\aleph_1$ dense subsets of reals, but just 1 for $\aleph_1$ dense subsets of reals closed under Turing equivalence.

Spring 2018

January 29, Organizational meeting

This day we decided the schedule for the semester.

February 5, Uri Andrews

Title: Building Models of Strongly Minimal Theories - Part 1

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 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

Title: Finding Definable Sets in Continuous Logic

Abstract: In order to be useful the notion of a 'definable set' in continuous logic is stricter than a naive comparison to discrete logic 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, Noah Schweber

Title: Proper forcing

Abstract: Although a given forcing notion may have nice properties on its own, those properties might vanish when we apply it repeatedly. 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

Title: A survey of computable and constructive mathematics in economic history

March 5, Tamvana Makulumi

Title: Convexly Orderable Groups

March 12, Dan Turetsky (University of Notre Dame)

Title: Structural Jump

March 19, Ethan McCarthy

Title: Networks and degrees of points in non-second countable spaces

April 2, Wil Cocke

Title: Characterizing Finite Nilpotent Groups via Word Maps

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.

April 9, Tejas Bhojraj

Title: Quantum Randomness

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.

April 16, Iván Ongay-Valverde

Title: What can we say about sets made by the union of Turing equivalence classes?

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?

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.

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

April 23, Ethan McCarthy (Thesis Defense) Start 3:45 Room B231

Title: Cototal enumeration degrees and their applications to effective mathematics

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.

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]\displaystyle{ \omega^{\lt \omega} }[/math], and as the degrees of enumeration-pointed trees on [math]\displaystyle{ 2^{\lt \omega} }[/math], and we will remark on some additional applications of these characterizations.

April 30, Iván Ongay-Valverde

Title: Definibility of the Frobenius orbits and an application to sets of rational distances.

Abstract: In this talk I'll present a paper by Hector Pastén. We will talk about how having a formula that identify a Frobenius orbits can help you show an analogue case of Hilbert's tenth problem (the one asking for an algorithm that tells you if a diophantine equation is solvable or not).

Finally, if time permits, we will do an application that solves the existence of a dense set in the plane with rational distances, assuming some form of the ABC conjecture. This last question was propose by Erdös and Ulam.

Fall 2017

September 11, Organizational meeting

This day we decided the schedule for the semester.

September 18, Noah Schweber

Title: The Kunen inconsistency

Abstract: While early large cardinal axioms were usually defined combinatorially - e.g., cardinals satisfying a version of Ramsey's 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.

September 25, Noah Schweber

Title: Hindman's theorem via ultrafilters

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

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.

October 9, James Hanson

Title: Morley rank and stability in continuous logic

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 > 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, Tamvana Makulumi

Title: Boxy sets in ordered convexly-orderable structures.

October 30, Iván Ongay-Valverde

Title: Dancing SCCA and other Coloring Axioms

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.

November 6, Wil Cocke

Title: Two new characterizations of nilpotent groups

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.

Or...

Title: Centralizing Propagating Properties of Groups

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.

November 13, Steffen Lempp

Title: The computational complexity of properties of finitely presented groups

Abstract: I will survey index set complexity results on finitely presented groups.

November 20, Ethan McCarthy

Title: Strong Difference Randomness

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.

December 4, Tejas Bhojraj

Title: Quantum Algorithmic Randomness

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.

December 11, Grigory Terlov

Title: The Logic of Erdős–Rényi Graphs