Abstract. We will give a self-contained proof of a local rigidity property of the usual action of SL_2(Z) on R^2, and use this to obtain new information concerning the Borel reducibility hierarchy just beyond the (measure) hyperfinite equivalence relations.

Abstract. There are many similarities between the notions of weak equivalence and orbit equivalence of measure preserving actions of a countably infinite group. For instance, when the group is amenable then all of its free ergodic actions are orbit equivalent (Ornstein-Weiss) and also weakly equivalent (Foreman-Weiss). For non-amenable groups there are continuum many free ergodic actions up to orbit equivalence (Epstein), and it is known the same holds for weak equivalence for many non-amenable groups (Abért-Elek). Abért and Elek exhibited a weak containment rigidity phenomenon for strongly ergodic profinite actions and, prompted by the orbit equivalence superrigidity results of Popa, asked whether a free action could exhibit weak equivalence superrigidity in the sense that its weak equivalence class and isomorphism class coincide. In this talk I will present the (negative) answer to this question, showing that the situation for weak equivalence is in fact quite the opposite from orbit equivalence: for every free action b of any countably infinite group the set of isomorphism classes of actions weakly equivalent to b does not admit classification by countable structures.

Abstract. Roughly speaking, Herbrand's theorem states that in a universal theory with quantifier elimination, given any formula f(x,y), there are finitely many terms t_1(x),...,t_n(x) such that whenever f(a,y) is true for some y, then f(a,t_i(a)) is true for one of the terms t_i. We will present an "approximate version" of this theorem for continuous logic and show how it implies a theorem describing definable functions in Sigma_2 theories. We will present several applications of this result, both in classical logic and continuous logic.

Abstract. Different generalizations are obtained, of a known theorem by Kechris, saying that any $\Sigma^1_1$ set of the Baire space either is effectively sigma-bounded (that is, covered by a countable union of compact $\Sigma^1_1$ sets), or it contains a superperfect subset, in particular,

1) with covering by compact sets and equivalence classes of a given finite collection of $\Delta^1_1$ equivalence relations,

2) generalizations to $\Sigma^1_2$ sets,

3) generalizations true in the Solovay model.

A generalization to $\Sigma^1_1$ sets is obtained as well, of a theorem by Louveau saying that any $\Delta^1_1$ set of the Baire space either is effectively sigma-compact (equal to a countable union of compact $\Delta^1_1$ sets), or it contains a relatively closed superperfect subset.

Abstract. This will be a gentle introduction to Hrushovski amalgamations and some variants. This talk should provide more detail for the results that I will talk about in the colloquium as well as presenting a result about recursive spectra of strongly minimal theories in finite languages.

Abstract. We will prove a representation theorem for the dominating reals in the Hechler extension. Specifically, we will prove that any dominating real eventually dominates the 'sandwich' composition of the Hechler real with two ground model reals that limit to infinity. Using this we will find an unbounded real in the Hechler extension which is dominated by every dominating real.

Abstract. Hechler forcing is the most basic tool for adding a dominating real to the universe, and several variations occur in the literature. We present results exploring the relationship between dominating and unbounded reals in these forcing extensions. We will show that in the tree Hechler extension every unbounded real is infinitely often above some dominating real, while in the standard Hechler extension there is an unbounded real eventually below every dominating real. As a corollary we negatively settle a conjecture of Brendle and Loewe. Our techniques also allow us to give a simple representation theorem for the dominating reals in the standard Hechler extension.

Abstract. We introduce diagonal property and use it to obtain Ramsey classes of structures related to finite ultrametric spaces, posets and linear orderings. In addition to this we give an application of our results to calculus of universal minimal flows.

Abstract. We consider Borel actions of a countable group G on a standard Borel space X and investigate situations when X admits a finite generator, equivalently, when there is a Borel G-embedding of X into k^G, for some finite k. One of our results states that if X is aperiodic (i.e. all orbits are infinite) then it admits a 4-generator modulo a meager set, answering a question raised by Kechris in the 90s.

Finite generators naturally arose in the entropy theory of measure preserving transformations on standard probability spaces. The Kolmogorov-Sinai theorem implies that if there is an invariant probability measure with infinite entropy then X doesn't admit a finite generator. In the 80s, Weiss asked whether the non-existence of an invariant probability measure implies existence of a finite generator. We prove a result towards this question, a corollary of which is that for all G the answer to Weiss's question is positive when X is a sigma-compact subset of a Polish space (and G acts continuously).

Abstract. In classical logic, an L-structure M is said to be pseudofinite if every L-sentence which is true in all finite L-structures is also true in M; equivalently, if an L-sentence is true in M, then it is true in some finite L-structure. The random graph is a pseudofinite structure and pseudofinite fields have proven to be very interesting to model theorists. In joint work with Vinicius Cifu Lopes, we initiate the study of pseudofinite metric structures (in the sense of continuous logic). Due to the lack of negations in continuous logic, the aforementioned equivalence doesn't hold, leading to two separate notions, which we call pseudofinite and strongly pseudofinite. By replacing finite structures by compact structures, we obtain the related notions of pseudocompact and strongly pseudocompact. In this talk, I will discuss some basic properties of these notions as well as many examples. I will also discuss some interesting open questions.

Abstract. The study of the dynamics of automorphism groups of countable homogenous structures (such as the random graph or the rational order) over the last decade or so has lead to some interesting connections between diverse areas of mathematics, including model theory, descriptive set theory, finite combinatorics (especially Ramsey theory), metric geometry, topological dynamics, ergodic theory, group representation theory and asymptotic geometric analysis. In this talk I will give a survey, aimed at a general mathematical audience, of recent developments in this area.

Abstract. Given a sequence of standard measure spaces and a non-principal ultrafilter U on the natural numbers, the ultraproduct of these spaces with respect to U is defined, using the Loeb measure construction. I will define on this space the ultraproduct action associated with a sequence of measure preserving actions of a countable group and I will discuss several applications to graph combinatorics of group actions and an application to sofic actions.

Abstract. In 2004, Zilber constructed a class of exponential fields, known as pseudoexponential fields, and proved that there is exactly one pseudoexponential field in every uncountable cardinality up to isomorphism. He conjectured that the pseudoexponential field of size continuum, K, is isomorphic to the classic complex exponential field. Since the complex exponential field contains the real exponential field, one consequence of this conjecture is the existence of a real closed exponential subfield of K. In this talk, I will prove the existence of uncountably many non-isomorphic countable real closed exponential subfields of K.

Abstract. Let H be an infinite-dimensional (real or complex) Hilbert space, viewed as a metric structure in the natural continuous signature for Hilbert spaces. We characterize the definable linear operators on H as exactly the "scalar plus compact" operators on H. We will also see how this shows that many naturally occurring operators from functional analysis are not definable in the sense of continuous logic.

Abstract. Horner's rule is optimal for polynomial 0-testing Abstract: The value V_{F,n}(a_0,...,a_n,b)=a_0+a_1b+...+a_nb^n of a polynomial of degree n>0 over a field F can be computed by Horner's rule using no more than n multiplications in F, and it is optimal (for many fields, including the reals and the complexes) for the number of multiplications and/or divisions required by a classical result of Pan. I will prove in outline that it is similarly optimal for polynomial 0-testing, the decision problem for the relation N_{F,n}(a_0,...,a_n,b)<==> a_0+a_1b+...+a_nb^n=0. The result holds for all usual computation models and (arguably) for all algorithms which decide N_{F,n}(a_0,...,a_n,b) from the field primitives 0,1,+,-,x,\div and =.

Abstract. The tree property at $\kappa^+$ states that every tree with height $\kappa^+$ and levels of size at most $\kappa$ has an unbounded branch. We will present some consistency results about the tree property at small cardinals and the relationship between the tree property and the singular cardinal hypothesis.

Abstract. We will outline some motivations that lead to the study of fine structural models otherwise known as mice. Time permitting, we will discuss some current developments.

Abstract. One of the major projects of proof theory is extracting explicit bounds from suitable non-effective proofs. The main method for doing this, the functional interpretation, depends heavily on syntactic properties of the statements, and have traditionally been limited to proofs $\Pi_2$ statements. However new variations of the functional interpretation make it possible to extract "bounds" from statements which are "relatively $\Pi_2$". We'll discuss how these methods can be applied to questions about computability and reverse mathematics in non-effective proofs of $\Pi1_2$ statements.

Abstract. I will present an overview of the ergodic theory of groups acting quasi-invariantly on probability spaces (meaning the measure is not preserved by the action but the null sets are). Such actions arise naturally in the context of Lie groups acting on symmetric space and automorphims of trees acting on graphs. The bulk of the talk will be background and introductory material; I will conclude with a description of my own research and results in this area.

Abstract. In 1931, motivated by physical examples, von Neumann proposed classifying the ergodic measure preserving transformations of standard probability spaces. Much recent work has used the tools of descriptive set theory to show that this is impossible for abstract measure preserving transformations.

This talk, which discusses joint work with B. Weiss, extends the anticlassification theorems to the space of measure preserving diffeomorphisms of the 2-torus.

Abstract. In 1931, motivated by physical examples, von Neumann proposed classifying the ergodic measure preserving transformations of standard probability spaces. Much recent work has used the tools of descriptive set theory to show that this is impossible for abstract measure preserving transformations.

This talk, which discusses joint work with B. Weiss, extends the anticlassification theorems to the space of measure preserving diffeomorphisms of the 2-torus.

Abstract. An important geometric invariant of a finitely generated group is its space of ends. The space of ends of an arbitrary topological space may be intuitively described as the set of "path components at infinity." For proper geodesic spaces, I show how to use the language of nonstandard analysis to make the aforementioned heuristic precise. When this nonstandard characterization is applied to the case of a Cayley graph of a finitely generated group, it becomes easier to perform calculations and prove theorems, as will be illustrated through a few examples. I will end the talk with some ideas for future applications.

Abstract. The Borel chromatic number of a graph on a Polish space is the least number of colors needed to color its vertices by a Borel function. Kechris-Solecki-Todorcevic (1999) isolate a graph G_0 which is minimal among graphs with uncountable Borel chromatic number, and ask whether there is an analogous minimal object among graphs of infinite Borel chromatic number. We investigate this question and some refinements. This is joint work with B.D. Miller and A.S. Kechris.