Colloquium: Itay Glazer (Northwestern)

Thu, 17/11/202214:30-15:30
Title: Word maps and word measures: probability and geometry

Abstract: Every word w(x_1,...,x_r) in a free group, such as the commutator word w=xyx^(-1)y^(-1), induces a word map w:G^r-->G on every group G. For g in G, it is natural to ask whether the equation w(x_1,...,x_r)=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), one can study the geometry of the polynomial map w:SLn(C)^r-->SLn(C), using algebraic methods.Such problems have been studied in the last few decades, in various settings such as finite simple groups, compact p-adic groups, compact Lie groups, and simple algebraic 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, Raf Cluckers and Nir Avni.

Recording will be available at: