Activities This Week

Recurrence is a classical topic in ergodic theory and dynamical systems, which goes back to Poincaré’s recurrence theorem. I will talk about old, less old, and new results on recurrence. In particular, I will talk about how to obtain asymptotic results on the number of times a typical point returns to a shrinking neighbourhood around itself.

Rokhlin dimension is a regularity property for group actions on $C^*$-algebras. It was originally introduced for actions of the integers and finite groups, and later the definition was extended to other classes of groups. Rokhlin dimension comes in two flavors, commuting and non-commuting towers, which at least for finite group actions, turn out to be different. The main interest in Rokhlin dimension was as a tool to show that various regularity properties of a $C^*$-algebra pass to the crossed product. For those types of theorems, one only cares about whether this dimension is finite or infinite, and not the actual value. For actions of finite groups on simple $C^*$-algebras, the only known examples had dimensions 0,1,2 or infinity. Nuclear dimension, a related non-dynamical dimension for $C^*$-algebras, is known to only admit the values 0,1 or infinity on simple $C^*$-algebras, so it might seem plausible that Rokhlin dimension would exhibit similar behavior. In this talk, I’ll describe work in preparation which shows that arbitrarily large values can be achieved (though we don’t know how to achieve all known examples), as well as finer conclusions which can be deduced from the actual value, as opposed to merely whether the dimension is finite. This shows that the value Rokhlin dimension can in fact be seen as an interesting invariant of the group action. The tools required for proving it involve equivariant K-theory and the Atiyah-Segal completion theorem; I will not assume that the audience is familiar with those.

This is joint work with N. Christopher Phillips.

Let f:X \to Y be a morphism between smooth, geometrically irreducible Z-schemes of finite type. We study the number of solutions #{x:f(x)=y mod p^k} for prime p, positive number k, and y \in Y(Z/p^kZ), and show that the geometry and singularities of the fibers of f determine the asymptotic behavior of this quantity as p, k and y vary.

In particular, we show that f:X \to Y is flat with fibers of rational singularities, a property abbreviated (FRS), if and only if #{x:f(x)=y mod p^k}/p^{k(\dim X -\dim Y)} is uniformly bounded in p, k and y. We then consider a natural family of singularity properties, which are variants of the (FRS) property, and provide for each member of this family a number theoretic characterization using the asymptotics of #{x:f(x)=y mod p^k}/p^{k(\dim X -\dim Y)}.

To prove our results, we use model theoretic tools (and in particular the theory of motivic integration, in the sense of uniform p-adic integration) to effectively study the collection {#{x:f(x)=y mod p^k}/p^{k(\dim X -\dim Y)}. If time allows, we will discuss these methods.

Based on a joint work with Raf Cluckers and Itay Glazer.

Given a word w in a free group F_r on a set of r elements (e.g. the commutator word w=xyx^(-1)y^(-1)), and a group G, one can associate a word map w:G^r–>G. For g in G, it is natural to ask whether the equation w(x1,…,xr)=g has a solution in G^r, and to estimate the “size” of this solution set, in a suitable sense. When G is finite, or more generally a compact group, this becomes a probabilistic problem of analyzing the distribution of w(x_1,…,x_r), for Haar-random elements x_1,…,x_r in G. When G is an algebraic group, such as SLn(C), it is natural to study the geometry of the fibers of w. Such problems have been extensively studied in the last few decades, in various settings such as finite simple groups, compact p-adic groups, compact Lie groups, simple algebraic groups, and arithmetic groups. Analogous problems have been studied for Lie algebra word maps as well. In this talk, I will mention some of these results, and explain the tight connections between the probabilistic and algebraic approaches.

Based on joint works with Yotam Hendel and Nir Avni.