A compact space is a Rosenthal compactum if it can be embedded into the space of Baire class 1 functions on a Polish space. Those objects have been well studied in functional analysis and set theory. In this talk, I will explain the link between them and the model-theoretic notion of NIP and how they can be used to prove new results in model theory on the topology of the space of types.

Hide abstract

Infinite games with Borel payoff are determined, and this cannot be proven without appeal to some form of the axioms of Replacement and Power set. But what ambient set theory is strictly necessary? In this talk, we isolate a family of reflection principles that correspond exactly in strength to determinacy for certain levels of the Borel hierarchy. We also discuss some connections with measurable cardinals and higher order reverse mathematics.

Hide abstract

In the last several years there have been new methods introduced into the theory of countable Borel equivalence relations which have solved some problems which were previously not accessible. These new methods center around an interplay between forcing arguments and the combinatorial structure of the equivalence relation. We will present a number of results along these lines, including some very recent results. Interestingly, some of these results for Z^d seem to be analogs of results by Marks for the free groups, but the techniques seem to be completely different.

Hide abstract

It is known that in $o$-minimal structures, any unbounded definable curve inside a definable group $G$ gives rise to a one-dimensional torsion-free definable subgroup of $G$, associated to the curve at ``infinity'' (a result of Peterzil and Steinhorn, 1999). In this talk I will recall this construction and describe a generalization of the result to definable types of arbitrary dimension: Each definable type $p$ inside $G$ gives rise to a definable, torsion-free subgroup $H_p$, whose dimension can be read-off the type $p$.

As we will see, the group $H_p$ is the stabilizer of a partial type associated to $p$, under the action of $G$ on a certain quotient of the usual types space $S(G)$. It turns out that this quotient of the type space is closely related to the classical construction of Samuel’s compactification of topological groups and more generally of spaces with uniformity.

Hide abstract

The finite sharply 2- and 3-transitive groups were classified by Zassenhaus in the 1930 and all (but finitely many) arise (essentially) from fields. It was an open question whether the same is true in the infinite case. We answer this question.

Hide abstract

A major theme in ergodic Ramsey theory is proving multiple recurrence results for certain doubly recurrent (mixing) actions of semigroups. The amplification of double to multiple recurrence is usually done using a so-called van der Corput difference lemma for a suitable filter on the semigroup. Particular instances of this lemma (for concrete filters) have been proven before (by Furstenberg, Bergelson-McCutcheon, and others), with a different proof for each filter. We define a notion of differentiation for subsets of semigroups and isolate the class of filters that respect this notion. The filters in this class (call them Delta-filters) include all of those for which the van der Corput lemma was known, and our main result about them is a Ramsey theorem related to labeling edges between the semigroup elements with their ratios. An application of this theorem yields a van der Corput lemma for Delta-filters, generalizing all its previous instances.

Hide abstract