Date:

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:**

__https://huji.cloud.panopto.eu/Panopto/Pages/Sessions/List.aspx?folderID=a6e6beab-eb0e-47c6-9d5d-aeb500acb5ad__