Abstract: If X is a set then (X choose a) is all a-subsets of X.
By Ramsey's theorem if you finitely color (N choose a) then there will be an infinite homog set H (that is, a set H such that the coloring on (H choose a) is constant-- just ONE color).
Let L be an ordering (for this talk Z or omega^2). What if you color (L choose a) and want, a homog set OF THE SAME ORDER TYPE AS L. We call that L-homog.
Alas- that almost never happens. You can color (Z choose 2) and have NO Z-homog set. However, for every finite coloring of (Z choose 2) there IS what we call a 4-Z-homog set- Only FOUR colors. More generally, for every finite coloring of (Z choose a) there is a 2^a-Z-homog set. And this is tight.
We will prove results about coloring (L choose a) for a variety of linear orders L and a\in N.
This work is joint with Joanna Boyland, Nathan Hurtig, Robert Rust.
Abstract: We will discuss how to define coarse properness of general topological groups on locally compact spaces and show how the existence of cocompact such actions provide a characterisation of bounded geometry of the group. We will also show how this may equivalently be defined for actions on commutative C*-algebras and discuss problems arising in the non-commutative case.
Abstract: We will present a number of problems and results in computable model theory, where we apply ideas, notions, and methods from computability theory to study algorithmic phenomena in countable algebraic structures. We will address complexity of structures, of their isomorphisms, and of additional relations on structures. Classically isomorphic structures can have very different computability-theoretic properties. When a complexity bound for some property of a structure is preserved under isomorphisms, we call the bound intrinsic. Often, we can describe complexity of an aspect of a structure syntactically – typically using computable infinitary formulas, or measure it semantically – typically using Turing degrees. This connection between definability and computability has been one of the main themes in computable model theory.
Abstract: The Laver tables are finite left self-distributive algebras on one generator which were first studied by Laver in the course of analyzing rank-to-rank elementary embeddings. Finite rooted ordered binary trees naturally define elements in these tables. I will show that, for each 0 < p < 1, measure 1 sets of (typically infinite) rooted ordered binary trees can also be evaluated in the Laver tables. This ad hoc result connecting two disparate subjects arose from the following consideration: If Thompson's group F were amenable, what mechanism is responsible for its amenability?
Abstract: After giving an overview of the basic properties of definable sets in continuous logic, we will give a largely visual proof that any finite upper semilattice is the partial order of definable sets in some superstable continuous first-order theory.
Abstract: I will describe some novel approaches to investigating the combinatorial topology of surfaces through model theoretic means. The main object of interest is the curve graph of a surface, which encodes the homotopy classes of essential embedded loops on the surface, and its automorphism group, which coincides with the extended mapping class group of the surface. I will give a model theoretic perspective on how a myriad of objects that are naturally associated to a surface are interpretable inside of the curve graph, and how this makes precise a certain metaconjecture due to Ivanov. I will also discuss some of the properties of the theory of the curve graph, including stability and quantifier elimination. This talk represents joint work with V. Disarlo and J. de la Nuez Gonzalez.
Abstract: In this talk, I will define what it means for a coloring of substructures of an ultraproduct structure to be ``internal’’, and a notion of finite big Ramsey degree for internal colorings. I will also present a certain Ramsey degree transfer theorem from countable sequences of finite structures to their ultraproducts, assuming AC and some additional mild assumptions. The big Ramsey degree of a finite structure in an ultraproduct can differ markedly from its internal big Ramsey degree, as demonstrated by the example of the class of all finite linear orders, which I will explain.
This is joint work with Dana Bartošová, Mirna Džamonja and Rehana Patel.
Abstract: A theory T is called binary if any two tuples have the same type if and only if all corresponding subtuples of length 2 have the same type. With this strong restriction on theories, it turns out that certain classification-theoretic dividing lines collapse: for example, we show that a binary NSOP_1 theory is simple, and a binary NSOP_3 theory is NTP_1. Motivated by these results, we develop the basics of neostability theory for the broader category of treeless theories. We show such theories come equipped with a natural notion of independence, defined in terms of generically stable partial types, which is meaningful in both simple and NIP theories. This is joint work with Itay Kaplan and Pierre Simon.
Abstract: In this talk, I will discuss various notions of indivisibility for (relational) structures. A structure is indivisible if, when you color the elements of the universe with finitely many colors, you can find a monochromatic copy of the structure. I will examine which structures have various levels of indivisibility and which do not, as well as general mechanisms for exhibiting indivisibility and non-indivisibility. Some of the work presented is joint with Miriam Parnes and Lynn Scow.
Abstract: In a series of lectures at Princeton University between 1935 and 1937, John von Neumann developed a continuous version of projective geometry: the central objects of this study, continuous geometries, are complete, complemented, modular lattices whose algebraic operations possess certain natural continuity properties. Beyond classical finite-dimensional projective geometries, the class of continuous geometries contains, for instance, every orthocomplemented complete modular lattice (e.g., the projection lattice of any finite von Neumann algebra).
In the course of his analysis, von Neumann established the following remarkable coordinatization theorem: every complemented modular lattice (with an order at least four) is isomorphic to the lattice of principal left ideals of some (up to isomorphism unique) regular ring. Furthermore, he proved that every irreducible continuous geometry possesses a unique dimension function (with values in the closed real unit interval), which then induces a compatible complete metric on the corresponding coordinatizing ring and thus furnishes the latter with a natural topology. The topological groups of units of such ”continuous rings” exhibit very peculiar dynamical behavior.
In the talk, I will give a brief overview of von Neumann’s continuous geometry and report on some recent progress in understanding the structure and dynamics of topological groups of units of continuous rings.
Abstract: I describe a project in countable model theory, namely the classification of first order theories according to their Borel complexity. The maximally bad theories are called "Borel Complete," and we expect that theories which are not Borel complete admit some sort of structure theory for countable models. The issue is that we have gotten stuck on several concrete theories; that is, we cannot figure out how to place them in the hierarchy. I describe these examples. These include the class of modules over certain highly noncommutative rings, and also various weakly minimal, trivial, theories, and also various weakly minimal, unidimensional theories. These last examples seem to be connected to the representation theory of quivers. Joint with Chris Laskowski.
Abstract: I will describe a surprisingly successful approach to repairing unrestricted comprehension motivated by ideas and techniques from continuous logic.
Abstract: The classical Banach-Mazur theorem states that every separable Banach space admits a linear isometric embedding into C[0,1], the separable Banach space of continuous functions on the unit interval, and we say that C[0,1] is an isometric universal space for the class of separable Banach spaces. In 1968, Szlenk showed that there is no isomorphic separable reflexive Banach space for the class SR of separable reflexive Banach spaces. Szlenk's result was significantly improved by Bourgain in 1980 when he showed that if a Banach space is isomorphic universal for the class SR then it must contain an isomorphic copy of C[0,1]. Bourgain's influential descriptive set theoretic argument relies on a tree ordinal index. In this talk, we introduce nonlinear indices in the spirit of Bourgain's tree indices and show some universality results in the Lipschitz and coarse category. While our Lipschitz universality result is valid in ZFC, one of the coarse universality results requires some additional set-theoretic axioms. This talk is based on a joint work with G. Lancien (Université Bourgogne Franche-Comté), P. Motakis (York University), and Th. Schlumprecht (Texas A&M University)
Abstract: Recently, Algom-Kfir and Bestvina introduced mapping class groups of locally finite graphs as a proposed analog of Out(F_n) in the infinite-type setting. Using a framework established by Rosendal, we will then discuss the coarse geometry of the pure mapping class groups and related properties, including results on asymptotic dimension. This is joint work with George Domat and Sanghoon Kwak.
Abstract: For the last 15 years, I have been interested in whether or not categoricity of L(omega1,omega) is preserved by forcing. The question can be phrased in terms of classes of atomic models of a complete, first order theory. The new result is that if an atomic class has a model of size larger than the continuum, then aleph1-categoricity implies omega-stability. This is joint work with Baldwin and Shelah.
Abstract: We study the Constraint Satisfaction Problems of unary expansion of (Z, succ) and the class of "alpha-hereditarily sparse" finite graphs forbidding triangles. We discuss some progress towards each direction.
Abstract: In this talk, we will discuss the framework of partial actions for constructing orbit equivalent actions of Polish groups. While related ideas have been employed in ergodic theory and Borel dynamics for many years, the particular viewpoint of partial actions simplifies construction of orbit equivalent actions of distinct groups.
As an application, we will present a Borel version of Katok's representation theorem for multidimensional Borel flows. One-dimensional flows are closely connected to actions of $\mathbb{Z}$ via the so-called "flow under a function" construction. This appealing geometric picture does not generalize to higher dimensions. Within the ergodic theoretical framework, Katok introduced the concept of a special flow as a way to connect multidimensional $\mathbb{R}^d$ and $\mathh{Z}^d$ actions. We will show that similar connections continue to hold in Borel dynamics.
Another illustration of the partial actions techniques that we intend to touch is the following result: a Borel equivalence relation generated by a free $\mathbb{R}$-flow can also be generated by a free action of any non-discrete and non-compact Polish group. This is in contrast with the situation for discrete groups, where amenability distinguishes groups that can and cannot generate free finite measure-preserving hyperfinite actions.
Abstract: As consequence of the VC theorem, any pseudofinite measure over an NIP ultraproduct is generically stable. We demonstrate a converse of this theorem and prove that finitely approximable measures over a pseudofinite structure are themselves pseudofinite (even without the NIP assumption). We then analyze the connection between the Morley product and the pseudofinite product. We show that if μ is definable and μ and ν are both pseudofinite, then the Morley product agrees with the pseudofinite product. Using this observation, we can construct generically stable idempotent measures on pseudofinite NIP groups. Our goal is to classify generically stable idempotents in this context.
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