Keisler measures were introduced in the late 80's by Keisler but they became central objects in model theory only recently with the development of NIP theories. This led naturally to the question of whether there might be a parallel theory of measures in other tame classes, especially in the simple theories where pseudofinite counting measures supply natural and interesting examples. We will describe some first steps toward establishing such a theory, based on Keisler randomizations and the theory of independence for NSOP1 theories in continuous logic.

We shall try to prove some surprising (and hopefully, correct) theorems about the relationship between the club principle (Hebrew: tiltan) and the splitting number, with respect to the classical s at omega and the generalized s at supercompact cardinals.

First speaker: Daniel kalmanovich, HU Title: On the face numbers of cubical polytopes Abstract: Understanding the possible face numbers of polytopes, and of subfamilies of interest, is a fundamental question. The celebrated g-theorem, conjectured by McMullen in 1971 and proved by Stanley (necessity) and by Billera and Lee (sufficiency) in 1980-81, characterizes the f-vectors of simplicial polytopes.

Expansivness is a fundamental property of dynamical systems. It is sometimes viewed as an indication to chaos. However, expansiveness also sets limitations on the complexity of a system. Ma\~{n}'{e} proved in the 1970’s that a compact metric space that admits an expansive homeomorphism is finite dimensional. In this talk we will discuss a recent extension of Ma\~{n}'{e}’s theorem for actions generated by multiple homeomorphisms, based on joint work with Masaki Tsukamoto. This extension relies on a notion called “topological mean dimension’’ , introduced by Gromov and

Given a set X, the notion of VC-dimension provides a way to measure randomness in collections of subsets of X. Specifically, the VC-dimension of a collection S of subsets of X is the largest integer d (if it exists) such that some d-element subset Y of X is ""shattered"" by S, meaning that every subset of Y can be obtained as the intersection of Y with some element of S. In this talk, we will focus on the case that X is a group G, and S is the collection of left translates of some fixed subset A of G.

The study of the representations theoretic properties of the group of diffeormorphisms of locally compact non compact Riemmanian manifolds which equal to the identity outside a compact set is is linked to a natural quasi invariant action of the group which moves all points of a Poisson point process according to the diffeomorphism (Gelfand-Graev-Vershik and Goldin et al.). Neretin noticed that the local diffeomorphism group is a subgroup of a larger group which he called GMS and that GMS also acts in a similar manner on the Poisson point process.

A major difficulty in finding polynomial patterns in primes is the need to understand their distribution properties at short scales. We describe how for some polynomial configurations one can overcome this problem by concatenating short scale behavior in "many directions" to long scale behavior for which tools from additive combinatorics are available.