Department of

Mathematics


Seminar Calendar
for Logic Seminar events the year of Saturday, October 19, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Friday, January 18, 2019

4:00 pm in 345 Altgeld Hall,Friday, January 18, 2019

Generic flat pregeometries

Omer Mermelstein (University of Wisconsin, Madison.)

Abstract: The property of "flatness" of a pregeometry (matroid) is best known in model theory as the device with which Hrushovski showed that his example refuting Zilber's conjecture does not interpret an infinite group. I will dedicate the first part of this talk to explaining what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. In the second part, I will conjecture that the family of flat pregeometries associated to strongly minimal sets is model theoretically nice, and share some intermediate results.

Tuesday, January 22, 2019

1:00 pm in Altgeld Hall,Tuesday, January 22, 2019

A Homotopical View of Lascar Groups of First-Order Theories

Greg Cousins (Notre Dame)

Abstract: In this talk, we will discuss how the Lascar group of a first-order theory, $T$, can be recovered as the fundamental group(-oid) of a certain space associated to the category of models, $Mod(T)$. We will then discuss some examples illustrating how tools from algebraic topology can be used to compute the Lascar group of a theory. Time permitting, we will discuss generalizations to the context of AECs and questions their higher homotopy. No knowledge of homotopy theory will be assumed. This is joint work with Tim Campion and Jinhe Ye.

Tuesday, January 29, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, January 29, 2019

Self-similar structures

Garret Ervin (Carnegie Mellon)

Abstract: An iterated function system is a finite collection $f_1, …, f_n$ of contraction mappings on a complete metric space. Every such system determines a unique compact subspace $X$, called the attractor of the system, such that $X = \bigcup f_i[X]$. Many well-known fractals, like the Cantor set and Sierpinski triangle, are realized as attractors of iterated function systems.
 A surprisingly rich analysis can be carried out even when the functions $f_i$ are only assumed to be non-surjective injections from a set to itself. Moreover, in many cases this analysis can be used to characterize when a structure $X$, like a group or linear order, is isomorphic to a product of itself, or to its own square. Such structures behave much like attractors of iterated function systems. We present the technique, and cite solutions to two old problems of Sierpinski as an application.

Tuesday, February 5, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, February 5, 2019

Local Keisler Measures and NIP Formulas

Kyle Gannon (Notre Dame)

Abstract: The connection between finitely additive probability measures and NIP theories was first noticed by Keisler. Around 20 years later, the work of Hrushovski, Peterzil, Pillay, and Simon greatly expanded this connection. Out of this research came the concept of generically stable measures. In the context of NIP theories, these particular measures exhibit stable behavior. In particular, Hrushovski, Pillay, and Simon demonstrated that generically stable measures admit a natural finite approximation. In this talk, we will discuss generically stable measures in the local setting. We will describe connections between these measures and concepts in functional analysis as well as show that this interpretation allows us to derive a local approximation theorem.

Tuesday, February 12, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, February 12, 2019

The Open Graph Dichotomy and the second level of the Borel hierarchy

Raphaël Carroy (Gödel Research Center for Math. Logic at Univ. of Vienna)

Abstract: I will explain how variants of the open graph dichotomy can be used to obtain various descriptive-set-theoretical dichotomies at the second level of the Borel hierarchy. This shows how to generalise these dichotomies from analytic metric spaces to separable metric spaces by working under the axiom of determinacy. If time allows it, I will also discuss some connections between cardinal invariants and the chromatic number of the graphs at stake.

Tuesday, February 19, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, February 19, 2019

Realizations of countable Borel equivalence relations

Forte Shinko (Caltech)

Abstract: By a classical result of Feldman and Moore, it is known that every countable Borel equivalence relation can be realized as the orbit equivalence relation of a continuous action of a countable group on a Polish space. However, if we impose further conditions, such as requiring the action to be minimal, then it is no longer clear if such a realization exists. We will detail the progress on characterizing when realizations exist under various conditions, including a complete description in the hyperfinite case. This is joint work with Alexander Kechris.

Tuesday, February 26, 2019

1:00 pm in Altgeld Hall,Tuesday, February 26, 2019

n-dependent groups and fields

Nadja Hempel (UCLA)

Abstract: NIP theories are the first class of the hierarchy of n-dependent structures. The random n-hypergraph is the canonical object which is n-dependent but not (n-1)-dependent. Thus the hierarchy is strict. But one might ask if there are any algebraic objects (groups, rings, fields) which are strictly n-dependent for every n? We will start by introducing the n-dependent hierarchy and present all known results on n-dependent groups and fields. These were (more or less) inspired by the above question.

Tuesday, March 5, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, March 5, 2019

Descriptive graph combinatorics with applications to geometry

Spencer Unger (Tel Aviv University)

Abstract: The Banach–Tarski paradox states that (assuming the axiom of choice) a unit ball in $\mathbb{R}^3$ can be partitioned into $5$ sets which can be rearranged by isometries to partition two unit balls. This famous result is part of a larger line of early 20th century research which sought to understand the relation between foundations of measure theory and generalizations of classical ideas such as decomposing polygons into congruent sets.
 In the last few years, there has been a resurgence of interest in these geometrical paradoxes. These results have the unifying theme that the "paradoxical" sets in many classical geometrical paradoxes can surprisingly be much "nicer" than one would naively expect. In this talk, we give a survey of these results and explain a few of the ideas that go in to a constructive solution to Tarski's circle squaring problem. This is joint work with Andrew Marks.

Tuesday, March 12, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, March 12, 2019

Hyperfiniteness and descriptive combinatorics

Clinton Conley (Carnegie Mellon)

Abstract: We survey some recent results on connections between descriptive set-theoretic properties of Borel graphs and hyperfiniteness of their connectedness equivalence relation. For convenience, we will focus on chromatic numbers with various measurability constraints. This talk will include joint work with Jackson, Marks, Miller, Seward, Tucker-Drob.

Tuesday, March 26, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, March 26, 2019

Cancelled

Friday, March 29, 2019

4:00 pm in 345 Altgeld Hall ,Friday, March 29, 2019

Generalized sum-product phenomenon for polynomials

Souktik Roy (UIUC Math)

Abstract: Suppose $P(x,y)$ and $Q(x,y)$ are real polynomials with non-trivial dependence on $x$ and $y$, and $\epsilon$ is any positive constant. If, for a sufficiently large $n$-element set $A$ of real numbers, both $|P(A,A)|$ and $|Q(A,A)|$ are simultaneously smaller than $n^{5/4-\epsilon}$, then we shall prove that either \[ P(x,y) = f(u(x)+Cu(y)) \text{ and } Q(x,y) = g(u(x)+Du(y)), \] or \[ P(x,y) = f(u(x)u^{c}(y)) \text{ and } Q(x,y) = g(u(x)u^{d}(y)), \] where $f,g,u$ are polynomials and $C,D,c,d$ are constants. As a corollary, we obtain a strengthening of a classic result of Elekes and Rónyai in a symmetric setting of natural interest. The proof combines ideas from incidence geometry and o-minimality in model theory. This is joint work with Yifan Jing (UIUC) and Minh Chieu Tran (UIUC).

Tuesday, April 9, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, April 9, 2019

Multiplication of weak equivalence classes

Anton Bernshteyn (Carnegie Mellon)

Abstract: The relations of weak containment and weak equivalence were introduced by Kechris in order to provide a convenient framework for describing global properties of p.m.p. actions of countable groups. Weak equivalence is a rather coarse relation, which makes it relatively well-behaved; in particular, the set of all weak equivalence classes of p.m.p. actions of a given countable group $\Gamma$ carries a natural compact metrizable topology. Nevertheless, a lot of useful information about an action (such as its cost, type, etc.) can be recovered from its weak equivalence class. In addition to the topology, the space of weak equivalence classes is equipped with a multiplication operation, induced by taking products of actions, and it is natural to wonder whether this multiplication operation is continuous. The answer is positive for amenable groups, as was shown by Burton, Kechris, and Tamuz. In this talk, we will explore what happens in the nonamenable case. Number theory will make an appearance.

Friday, April 12, 2019

4:00 pm in 345 Altgeld Hall ,Friday, April 12, 2019

Ultraproducts as a tool in the model theory of metric structures

Ward Henson (UIUC)

Abstract: L is a signature of continuous first order logic for metric structures and we have a class C of L-structures which we want to investigate from the point of view of model theory. In general, this involves letting T be the L-theory of C, and working to understand the models of T as fully as possible. This means not only knowing which L-structures are models of T, but also understanding the definable predicates and (especially important) the definable sets in models of T. (A valuable byproduct might be an explicit axiomatization of T.) In this talk we will lay out how understanding ultraproducts of members of C can be an important practical tool for understanding the full class of models of T. As much as time permits, we will discuss examples that have been successfully treated in this way, including some new ones, focusing on Banach spaces and Banach lattices. (Most of this work on examples is part of a collaboration with Yves Raynaud.)

Tuesday, April 16, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, April 16, 2019

Positive model theory and sober spaces

Levon Haykazyan (University of Waterloo)

Abstract: I will talk about positive model theory (also known as coherent logic) where formulas are not closed under negation. This setting is in fact more general that full first-order logic, since negation can be expressed by changing the language. The result is that we can have as much negation as necessary, however no extra negation is forced by the framework.
  We can associate to a positive theory a natural spaces of types, which will no longer be Hausdorff, but (quasi-)compact and sober. I will show that these spaces play the role of the Stone spaces in the full first-order logic. In particular I will show how classical results (due to Vaught) connecting the structure of countable models to Stone spaces carry over to the positive setting, provided we find the appropriate formulations of topological properties for non-Hausdorff spaces.

Tuesday, April 23, 2019

1:00 pm in 345 Altgeld Hall ,Tuesday, April 23, 2019

Expansions of the real field which does not introduce new smooth functions

Alex Savatovsky (Universität Konstanz)

Abstract: We will give some conditions under which an expansion of the real field does not define new smooth functions. We will give a very rough sketch of the proof and discuss generalizations.

Friday, April 26, 2019

4:00 pm in 345 Altgeld Hall ,Friday, April 26, 2019

On generic monothetic subgroups of Polish groups

Dakota Ihli (UIUC Math)

Abstract: Given a Polish group $G$, what can be said about the subgroup $\overline{\left\langle g \right\rangle}$ for the generic element $g \in G$? In this talk we will discuss progress and open problems in this area. Special emphasis will be given on the group of measure-preserving automorphisms of the unit interval.

Tuesday, April 30, 2019

4:00 pm in 2 Illini Hall,Tuesday, April 30, 2019

Amenable first order theories

Anand Pillay (Notre Dame Math)

Abstract: (Joint with Hrushovski and Krupinski.) We extract from the properties (extreme) amenability of automorphisms groups of omega-categorical theories, the notion of (extreme) amenability of a first order theory T, which is much less restrictive than the automorphism property. I discuss some results around the Lascar group of such a theory, some proofs of which use versions of continuous logic.

Tuesday, May 7, 2019

1:00 pm in 345 Altgeld Hall ,Tuesday, May 7, 2019

An Intuitive Approach to the Martin Boundary

Peter Loeb (UIUC Math)

Abstract: The talk uses Robinson’s nonstandard analysis to give a rigorous, but intuitive, probabilistic construction of a compactifying boundary with maximal representing measures for positive harmonic functions.

Tuesday, August 27, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, August 27, 2019

Organizational meeting

Tuesday, September 3, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, September 3, 2019

Ergodic theorems and more fun with countable Borel equivalence relations

Jenna Zomback (UIUC Math)

Abstract: I will discuss my current and future work, which lies in the study of countable Borel equivalence relations (CBERs) and its applications to ergodic theory and measured group theory. In the first section of the talk, I will discuss a tiling result for amenable groups along Tempelman Følner sequences and explain how this result implies the corresponding pointwise ergodic theorem (this is joint work with Jonathan Boretsky). In the second section, I will introduce the notion of cost of an equivalence relation, and state a few important results in this field. In each of the two sections, I will state some proposed avenues for future work. This talk is part of a preliminary examination.

Tuesday, September 10, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, September 10, 2019

An ergodic advertisement for descriptive graph combinatorics

Anush Tserunyan (UIUC Math)

Abstract: Dating back to Birkhoff, pointwise ergodic theorems for probability measure preserving (pmp) actions of countable groups are bridges between the global condition of ergodicity (measure-theoretic transitivity) and the local combinatorics of the actions. Each such action induces a Borel equivalence relation with countable classes and the study of these equivalence relations is a flourishing subject in modern descriptive set theory. Such an equivalence relation can also be viewed as the connectedness relation of a locally countable Borel graph. These strong connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, probability theory, and descriptive graph combinatorics. I will discuss how descriptive set theoretic thinking combined with combinatorial and measure-theoretic arguments yields a pointwise ergodic theorem for quasi-pmp locally countable graphs. This theorem is a general random version of pointwise ergodic theorems for group actions and is provably the best possible pointwise ergodic result for some of these actions.

Tuesday, September 17, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, September 17, 2019

Conjugacy classes of automorphism groups of linearly ordered structures

Aleksandra Kwiatkowska (Universität Münster and Uniwersytet Wrocławski)

Abstract: In the talk, we will address the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a countable structure or of a Fraisse limit) that is extremely amenable and has ample generics. In fact, it is unknown if there exists a linearly ordered structure whose automorphism group has a comeager $2$-dimensional diagonal conjugacy class.
  We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. Moreover, we provide general conditions implying that there is no comeager conjugacy class or comeager $2$-dimensional diagonal conjugacy class in the automorphism group of an ordered Fraisse limit.
  This is joint work with Maciej Malicki.

Tuesday, September 24, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, September 24, 2019

Speeds of hereditary properties and mutual algebricity

Caroline Terry (U Chicago Math)

Abstract: A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property $H$, the speed of $H$ is the function which sends an integer $n$ to the number of distinct elements in $H$ with underlying set $\{1,...,n\}$. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete "jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollobás, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss. This is joint work with Chris Laskowski.

Tuesday, October 1, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, October 1, 2019

Continuity of Functions on the Powerset of the First Uncountable Cardinal

William Chan (University of North Texas)

Abstract: Assume the axiom of determinacy. In this talk, we will discuss an almost everywhere uniformization for club subsets of $\omega_1$. Using this uniformization, we will show that every function $\Phi : [\omega_1]^{\omega_1} \rightarrow \omega_1$ has a club $C \subseteq \omega_1$ so that $\Phi | [C]^{\omega_1}$ is a continuous function. This continuity result can be used to make the following cardinality distinction, $|[\omega_1]^{<\omega_1}| < |[\omega_1]^{\omega_1}|$. This is joint work with Stephen Jackson.

Tuesday, October 8, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, October 8, 2019

A Categorical Semantics for Linear (and Quantum) Dependent Type Theory

Kohei Kishida (UIUC Philosophy)

Abstract: Desirable features of a quantum programming language include: treating both quantum resources and classical control; being a functional language admitting semantics and other formal methods; and dependent types. The type theory of such a language must be linear to reflect the linearity of quantum processes, and we want it to involve parameters (values that are known at circuit generation time) and states (values known at circuit execution time) to describe and generate quantum circuits. The goal of this talk is to provide a general semantic structure for linear dependent type theory of that sort. We review categorical models of quantum processes and the sets-and-functions model of classical dependent type theory, and show how they can be integrated to model linear dependent type theory of classical parameters and quantum states. This is joint work with Frank Fu, Julien Ross, and Peter Selinger.

Tuesday, October 22, 2019

1:00 pm in Altgeld Hall,Tuesday, October 22, 2019

Orbit equivalence relations of some classes of non-locally compact Polish groups

Joseph Zielinski

Abstract: By results of A.S. Kechris, whenever a locally compact Polish group acts continuously on a Polish space, the orbit equivalence relation of the action is essentially countable—that is, Borel reducible to the orbit equivalence relation of an action of a countable group. It is unknown if this characterizes the locally compact Polish groups. S. Solecki, after proving an analogous characterization for smooth actions of compact Polish groups, showed this to be true in the case where the group, G, is the additive group of a separable Banach space. The characterization also holds for abelian pro-countable groups, by results of M. Malicki. We discuss recent work on this problem, including an extension of this characterization to some important classes of Polish groups. This is joint work with A.S. Kechris, M. Malicki, and A. Panagiotopoulos.

Tuesday, November 5, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, November 5, 2019

Deciding the theories of expansions of the real ordered group via Ostrowski numeration

Christian Schulz (UIUC Math)

Abstract: For which irrational numbers $\alpha$ does the theory of $(\mathbb{R}, <, +, \mathbb{Z}, \alpha \mathbb{Z})$ have a decision algorithm? Previously, this was known for quadratic $\alpha$ thanks to work by Hieronymi. In this talk, we present the progress so far on generalizing this result to non-quadratic $\alpha$. We also discuss applications of this work to the study of combinatorics on words using automated theorem-proving software. We end with a discussion of potential future work and goals.

Tuesday, November 12, 2019

1:00 pm in 345 Altgeld Hall,Tuesday, November 12, 2019

Model theory of $\mathbb{R}$-trees and of ultrametric spaces

Ward Henson (UIUC Math)

Abstract: First, we consider the class of metric spaces $(M,d)$ that are $\mathbb{R}$-trees with a convex metric. To treat this class using continuous first order logic, we fix a base point $p$ in $M$ and require that $M$ have radius at most $r$ with respect to $p (r>0)$. The class of these structures $(M,d,p)$ is axiomatizable. Moreover, the theory of this class has a model companion $T$, whose models we describe precisely. This theory is a well behaved continuous theory. For example, $T$ has QE and is complete; it is stable (but not superstable) and has the maximum possible number of models in each infinite cardinal.
  Second, given a model $M = (M,d,p)$ of $T$, we consider the closed subset $E_r(M) := \{x \in M | d(p,x)=r\}$. This is a definable set for $ $T, and the entire structure $M$ can be reconstructed from $(E_r(M),d)$. The metric $d$ on $E_r(M)$ is an ultrametric; further, at every $x \in E_r(M)$, the set of distances $\{d(x,y) | y \in E_r(M)\}$ is dense in the interval $[0,2r]$. These properties are easily seen to be axiomatizable in continuous logic, and we let $T^*$ denote the resulting theory. We show that $T^*$ has QE, so it is complete; further, $T$ and $T^*$ are bi-interpretable.
  This is joint work with Sylvia Carlisle.