Activities This Week

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.

In this talk, I will describe joint work with Mike Jury and Rob Martin. The focus of this talk will be on noncommutative (NC) rational function, i.e., elements of the free skew field on d generators. Suppose such a function is bounded on all finite-dimensional row contractions. In that case, it admits an inner-outer factorization as elements of the free semigroup algebra (analogous to the classical factorization in function theory on the disc). Both the inner and outer functions are NC rational, as well. I will describe the theory behind this factorization and discuss how one obtains representations of the Cuntz algebra $\mathcal{O}_d$ from inner elements of the free semigroup algebra. I will show that from NC rational inners, one obtains the finitely-correlated representations introduced by Bratelli and Jorgensen. I will finish the discussion with some open questions.