Mathematics

First order expansions of the ordered real additive group

Erik Walsberg (UIUC Math)

Abstract: I will discuss joint work with Philipp Hieronymi on first order expansions of $(\mathbb{R},<,+)$. The definable subsets of $\mathbb{R}^n$ in the known natural model-theoretically tame expansions are geometrically tame objects. Many of these examples satisfy the essentially strongest geometric tameness assumption, o-minimality. In this talk I will discuss type A expansions. The assumption of being type A appears to be the broadest possible generalization of o-minimality for expansions of $(\mathbb{R},<,+)$ as some of the main o-minimal tools extend to this setting, and do not appear to extend further.

First order expansions of the ordered real additive group — Part II

Philipp Hieronymi (UIUC Math)

Abstract: This is a (hopefully self-contained) continuation of Erik's talk from Tuesday. I will discuss in more detail expansions of the ordered real additive group that are of type B (ie expansions of $(\mathbb{R},<,+)$ that define an order $(D,\prec)$ of order-type $\omega$ whose underlying set $D$ is somewhere dense in $\mathbb R$). In many (all?) type B structures 0-definability is equivalent to recognizablility by a Buechi automaton. Therefore results about the definable sets in type B structures imply results about sets recognizable by such automata. In this talk I will focus on our current research on continuous definable functions in type B structures, and I will discuss how it connects to results in computer science by Chaudhuri, Sankaranarayanan and Vardi.

Canceled

Topological dimension through the lens of Baire category

Anush Tserunyan (Illinois Math)

Abstract: In 1913, Brouwer proved that the topological dimension of $\mathbb{R}^n$ is $n$, which implies that there is no continuous injection of $\mathbb{R}^{n+1}$ into $\mathbb{R}^n$. More recently, Izzo and Li wondered if the last statement survives when the requirement of injectivity is relaxed to being injective on a large set. They showed in 2013 that the answer is negative when the largeness is measure-theoretic, i.e. when the continuous function is required to be injective only on a conull set. However, they conjectured that the answer should be positive for the notion of largeness provided by Baire category, namely: there does not exist a continuous function $\mathbb{R}^{n+1} \to \mathbb{R}^n$ that is injective on a comeager set. We will discuss this conjecture and its dramatic (for the speaker) resolution.

A Descriptive Set in Topological Dynamics

Robert Kaufman (Illinois Math)

Abstract: Let $X$ be a compact metric space and $H(X)$ the group of homeomorphisms of $X$, a Polish group. Then the orbit of $x \in X$ under $h \in H(X)$ (the two-sided orbit) has an obvious definition; its closure is called the orbit-closure of $x$. When all the orbit-closures are minimal then $h$ is called "sharp". (Every orbit-closure contains a minimal one). There are two main theorems.
A. The set $S(X)$ of sharp homeomorphisms is always co-analytic.
B. For a certain $X$, $S(X)$ is not Borel.
In the proof of B we need a variant of Hurewicz' theorem (1930) on the class of uncountable compact sets. This variant should be (but so far isn't) a consequence of Hurewicz' theorem. I'll say a very few words about a new method of proof.

Strong conceptual completeness for ℵ₀-categorical theories

Jesse Han (McMaster Math)

Abstract: Suppose we have some process to attach to every model of a first-order theory some (permutation) representation of its automorphism group, compatible with elementary embeddings. How can we tell if this is "definable", i.e. really just the points in all models of some imaginary sort of our theory?
In the '80s, Michael Makkai provided the following answer to this question: a functor $\mathrm{Mod}(T) \to \mathrm{Set}$ is definable if and only if it preserves all ultraproducts and all "formal comparison maps" between them, called ultramorphisms (generalizing e.g. the diagonal embedding into an ultrapower). This is known as strong conceptual completeness.
Any general framework which reconstructs theories from their categories of models should be considerably simplified for $\aleph_0$-categorical theories. Indeed, we show:
If $T$ is $\aleph_0$-categorical, then $X : \mathrm{Mod}(T) \to \mathrm{Set}$ is definable, i.e. isomorphic to ($M \mapsto \psi(M)$) for some formula $\psi \in T$, if and only if $X$ preserves ultraproducts and diagonal embeddings into ultrapowers. This means that all the preservation requirements for ultramorphisms, which a priori get unboundedly complicated, collapse to just diagonal embeddings when $T$ is $\aleph_0$-categorical. We show this definability criterion fails if we remove the $\aleph_0$-categoricity assumption, by constructing examples of theories and non-definable functors $\mathrm{Mod}(T) \to \mathrm{Set}$ which exhibit this.

Some aspects of simple proofs informed by history, ancient and modern

Victor Pambuccian (Arizona State University Math)

Abstract: We'll look at (i) the existence of a direct proof (even within intuitionistic logic) of the Steiner-Lehmus theorem, stating that a triangle with two congruent angle bisectors must be isosceles, (ii) the most primitive arithmetic capable of proving the irrationality of $\sqrt{2}$, going back to the early Pythagoreans, and why it cannot prove the irrationality of $\sqrt{17}$, the case Theodorus of Cyrene apparently stumbled upon, and (iii) the simplest proof for the fact that 30 is the greatest number all of whose totitives are prime.

Weak containment in ergodic theory and representation theory

Peter Burton (U Texas at Austin)

Abstract: The relation of weak containment for unitary representations of locally compact groups is a very useful tool in comparing such representations. Recently Kechris introduced an analogous definition of weak containment for measure-preserving actions of countable discrete groups. We will discuss the relationship between these concepts, and present a result showing that weak containment of measure-preserving actions is an essentially stronger notion than weak containment of the corresponding Koopman representations.

Finite versus infinite: An intricate shift

Yann Pequignot (UCLA)

Abstract: The Borel chromatic number — introduced by Kechris, Solecki, and Todorcevic (1999) — generalizes the chromatic number on finite graphs to definable graphs on topological spaces. While the $G_0$ dichotomy states that there exists a minimal graph with uncountable Borel chromatic number, it turns out that characterizing when a graph has infinite Borel chromatic number is far more intricate. Even in the case of graphs generated by a single function, our understanding is actually very poor.
 The Shift Graph on the space of infinite subsets of natural numbers is generated by the function that removes the minimum element. It is acyclic but has infinite Borel chromatic number. In 1999, Kechris, Solecki, and Todorcevic asked whether the Shift Graph is minimal among the graphs generated by a single Borel function that have infinite Borel chromatic number. I will explain why the answer is negative using a representation theorem for $\Sigma^1_2$ sets due to Marcone.

Polish groupoids and continuous logic

Ruiyuan (Ronnie) Chen (Caltech)

Abstract: It is well-known that every non-Archimedean Polish group is isomorphic as a topological group to the automorphism group of a countable structure, and analogously, that every Polish group is isomorphic to the automorphism group of a separable metric structure. We will present a generalization of this result: every open locally Polish groupoid admits a full and faithful Borel functor to the groupoid of metric L-structures on the Urysohn sphere, for some countable metric language L. This partially answers a question of Lupini. We will also discuss the analogous result in the non-Archimedean case.

Transcendence bases in ZF

Jindrich Zapletal (University of Florida Math)

Abstract: It is consistent with ZF+DC that the reals have a Hamel basis but the complex numbers have no transcendence basis. The proof relies on the difference between modularity of the two associated pre-geometries.

Interpolative fusions - preservation results

Alex Kruckman (IU Bloomington)

Abstract: Fix languages $L_1$ and $L_2$ with intersection $L_\cap$ and union $L_\cup$. An $L_\cup$ structure $M$ is interpolative if whenever $X_1$ is an $L_1$-definable set and $X_2$ is an $L_2$-definable set, $X_1$ and $X_2$ intersect in $M$ unless they are separated by an $L_\cap$-definable set. Examples of interpolative structures abound in model theory, but only recently did Minh Tran and Erik Walsberg begin studying the class in the abstract. When $T_1$ is an $L_1$ theory and $T_2$ is an $L_2$ theory, we are interested in the class of interpolative fusions: interpolative structures which are models of the union theory $T_\cup$. Putting aside the nontrivial question of whether this class is elementary (I will assume that it is, axiomatized by a theory $T^{*}$), I will explain how stability-theoretic assumptions on the base $L_\cap$-theory $T_\cap$ lead to preservation results of the form "If $T_1$ and $T_2$ both satisfy property $P$, then $T^{*}$ satisfies property $P$". This is joint work with Minh and Erik.

Colorings of finite subgraphs of the universal k-clique-free graphs

Natasha Dobrinen (University of Denver)

Abstract: It is a central question in the theory of homogeneous relational structures as to which structures have finite big Ramsey degrees. This question, of interest for several decades, has gained recent momentum as it was brought into focus by Kechris, Pestov, and Todorčević in 2005. An infinite structure $S$ is homogeneous if any isomorphism between two finitely generated substructures of $S$ can be extended to an automorphism of $S$. A homogeneous structure $S$ is said to have finite big Ramsey degrees if for each finite substructure $A$ of $S$, there is a number $n$, depending on $A$, such that any coloring of the copies of $A$ in $S$ into finitely many colors can be reduced down to no more than $n$ colors on some substructure $S'$ isomorphic to $S$. This is interesting not only as a Ramsey property for infinite structures, but also because of its implications for topological dynamics.
 Prior to work of the speaker, finite big Ramsey degrees had been proved for a handful of homogeneous structures: the rationals (Devlin 1979) the Rado graph (Sauer 2006), ultrametric spaces (Nguyen Van Thé 2008), and enriched versions of the rationals and related circular directed graphs (Laflamme, Nguyen Van Thé, and Sauer 2010). According to Nguyen Van Thé , "so far, the lack of tools to represent ultrahomogeneous structures is the major obstacle towards a better understanding of their infinite partition properties." We address this obstacle by providing new tools to represent the universal $k$-clique-free graphs and developing the necesshary Ramsey theory to deduce finite big Ramsey degrees. The methods developed seem robust enough that correct modifications should likely apply to a large class of homogeneous structures omitting some finite substructures.

Superrigidity and measure equivalence

Robin Tucker-Drob (Texas A&M)

Abstract: Measure equivalence is an equivalence relation on countable groups introduced by Gromov as a measure theoretic counterpart to the goemetric notion of quasi-isometry. In the first part of this talk I will give a brief introduction to measure equivalence. I will then discuss some new joint work with Lewis Bowen in which we show that the class B, of groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli shifts, is invariant under measure equivalence. As a consequence we show that any nonamenable lattice in a product of noncompact locally compact groups must belong to the class B. This also has implications for entropy: we introduce a new kind of entropy called weak Pinsker entropy, and show that equivalence relations generated by free measure preserving actions of groups in the class B completely "remember" the weak Pinsker entropy of the action.

Organizational meeting

Distality in Pairs

Travis Nell (UIUC)

Abstract: The notion of distality attempts to isolate the notion of a purely unstable behavior in an NIP theory. I will examine certain cases of expansions of o-minimal structures by a unary predicate. Each of these examples will be non-distal. In the case that the predicate is a proper, dense elementary substructure, I will characterize the distal types. In the case of the expansion of an ordered divisible abelian group and the predicate is a dense independent set, I will talk about how to find a distal expansion of this structure.

An introduction to continuous logic

Erik Walsberg (UIUC)

Abstract: Continuous logic will be introduced. Basic knowledge of first order logic and ultraproducts will be assumed.

Tame and wild structures with randomness

Minh Chieu Tran (UIUC)

Abstract: We consider structures built up from components interacting in a partially random fashion with one another. Initially, we were lead to the study of such structures in an attempt to answer a question by van den Dries, Hrushovski, and Kowalski on finding applications of number-theoretic character sum results in model theory. Ultimately, we realized that the underlying idea can be used to built a general frame work that allows us to understand many examples and phenomena of the area. (This talk contains results from joint works with Bhardwaj, Kruckman, and Walsberg).

Introduction to continuous logic, continued.

Erik Walsberg

Abstract: I will continue to introduce continuous logic. This time I will try to cover the syntax and some examples of continuous structures. I will assume basic knowledge of first order model theory, and that you came to my previous talk.

Descriptive locale theory

Ruiyuan Chen (UIUC)

Abstract: A locale is, informally, a topological space without an underlying set of points, with only an abstract lattice of ``open sets''. Various results in the literature suggest that locale theory behaves in many ways like a generalization of descriptive set theory with countability restrictions removed. This talk will introduce locale theory from a descriptive set-theoretic point of view, and survey some known and new results which are common to both contexts.

Pseudofinite groups, arithmetic regularity, and additive combinatorics

Gabe Conant (University of Notre Dame)

Abstract: I will report on joint work with Pillay and Terry on arithmetic regularity (a group theoretic analogue of Szemerédi regularity for graphs) for sets of bounded VC-dimension in finite groups, which is proved using a local version generic compact domination for NIP formulas in pseudofinite groups. I will then present more recent work on nonabelian versions of certain "inverse theorems" from additive combinatorics, which are proved using pseudofinite model theory, and can be used to give alternate proofs of NIP arithmetic regularity for certain classes of finite groups.

On the model theory of group actions on probability measure algebras

Ward Henson (UIUC (Emeritus))

Abstract: We treat such group actions using continuous model theory. For a finite or countable set S, let L_S be the continuous signature for probability measure algebras expanded by unary function symbols, one for each element of S. In this language, let T_S be the set of axioms for probability algebras (which we denote Pr) together with conditions expressing that each of the unary function symbols is interpreted by an automorphism of the algebra. If G is a group generated by S, we consider the extension of T_S obtained by adding a condition for each word w on S that represents the identity in G, asserting that the composition of unary functions corresponding to w is the identity; denote this theory by T_S(G). The main result to be discussed in this talk is that each T_S has a model companion T*_S, for which we give explicit axioms; this model companion is complete and has quantifier elimination. Its models consist of very particular actions on atomless probability algebras by the free group generated by S. Expressing and justifying our axioms for T*_S requires some information about the model theory of atomless probability algebras, which will be discussed in the first part of the talk. It is also true that when G is an amenable group, then T_S(G) has a model companion, which is very well behaved, but it will not be discussed much in this talk. (This is work in progress with Alex Berenstein; especially, we are aiming to understand the models of T*_S better, when |S|>1.)

From univalence to the fundamental group of the circle

Egbert Rijke (UIUC Math)

Abstract: We show how the univalence axiom can be used to construct the fundamental cover of the circle, which is then used to show that the loop space of the circle is equivalent to the integers.

Expansions of the real field by discrete subgroups of $Gl_n(\mathbb{C})$

Erik Walsberg (UIUC)

Abstract: Let $\Gamma$ be an infinite discrete subgroup of $Gl_n(\mathbb{C})$. Then either $(\mathbb{R},<,+,\cdot,\Gamma)$ is interdefinable with $(\mathbb{R},<,+,\cdot, \lambda^{\mathbb{Z}})$ for some $\lambda \in \mathbb{R}$, or $(\mathbb{R},<,+,\cdot,\Gamma)$ defines the set of integers. When $\Gamma$ is not virtually abelian, the second case holds.

Surreal Substructures

Vincent Bagayoko (Paris 7)

Abstract: We study the class No of surreal numbers equipped with its natural order and simplicity relation. A "surreal substructure" of No is a subclass that is isomorphic to No itself for the restricted order and simplicity relation. Such surreal substructures frequently arise when studying various operations on the surreals such as algebraic operations, exponentiation, or infinite summation. This makes it worthwhile to investigate the general properties of surreal substructures in their own right. We will give their general properties, provide many examples, and focus on two applications of surreal substructures: to produce fixed points of certain specific surreal functions and to define classes of surreal numbers defined as simplest in a convex class of No. We will mention their use in studying the hyperserial structure of No conjectured by van der Hoeven.

Logarithmic Hyperseries

Elliot Kaplan (UIUC)

Abstract: In a joint work with Lou van den Dries and Joris van der Hoeven, we constructed the field of Logarithmic Hyperseries. This is a proper class-sized ordered differential field which is also equipped with a logarithm and a composition. In this talk, I will briefly detail the construction of this field and indicate how the logarithm, composition, and derivation interact with each other. I will also indicate where this field fits among transseries, logarithmic transseries, and the surreal numbers.

The groups $\mathbb{Z}$ and $\mathbb{Q}$ with predicates for being square-free

Neer Bhardwaj (UIUC)

Abstract: I will begin with a brief survey of model-theoretic tameness properties of some of the most natural abelian groups. Then I move on to describe our work where we consider two expansions each of groups of integers and rational numbers with a predicate for being square-free. We prove that one of the structures is model theoretically wild while the other three structures are model theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences. This is joint work with Minh Chieu Tran.

Solution of Christensen's problem on universally measurable homomorphisms

Christian Rosendal (UIC Math)

Abstract: Answering a longstanding problem originating in J.P.R. Christensen’s seminal work on Haar null sets, we show that a universally measurable homomorphism between Polish groups is continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo ZF+DC, the existence of a discontinuous homomorphism between Polish groups implies that the Hamming graph on Cantor space has finite chromatic number.

Zero-one laws of Erdos–Renyi random graphs

Grigory Terlov (UIUC Math)

Abstract: This topic connects Probability theory with Logic. We will define a random graph, talk about why they are interesting in Probability theory, briefly touch on almost sure theories before spending majority of the time discussing various zero-one laws, including zero-one k-laws. This talk should be accessible to people without much experience in either subject.

Pairs of Theories Satisfying a Mordell-Lang Condition

Alexi Block Gorman (UIUC)

Abstract: In this talk I will discuss the expansion of geometric theories by a predicate that satisfies certain properties. Namely, the subset which the predicate defines is the universe of a model for some theory T' that interacts with the theory T of the larger structure in desirable ways. This framework generalizes the work of van den Dries on dense pairs of models of an o-minimal theory, the work of van den Dries and Gunaydin on pairs of the real or complex numbers with multiplicative subgroup satisfying the Mann property, and the work on lovely pairs of geometric structures developed by Berenstein and Vassilev. This talk is based on joint work with Philipp Hieronymi and Elliot Kaplan.