Title: On singularity properties of convolutions of algebraic morphisms and applications to probabilistic Waring-type problems
Speaker: Itay Glazer
Speaker Info: Weizmann Institute
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Special Note:
Abstract:
Let f and g be two morphisms from algebraic varieties X and Y to an algebraic group G. We define their convolution to be a morphism f∗g from X×Y to G by f∗g(x,y):=f(x)g(y). Similarly to the smoothing effect of the convolution operation in analysis, this operation yields morphisms with improved singularity properties. One motivation to consider this operation is the following; Aizenbud-Avni showed that for any semisimple algebraic group G, the commutator map from G×G to G, becomes flat with fibers of rational singularities (FRS), after 21 self-convolutions, providing applications to representation growth of compact p-adic groups and arithmetic groups. We generalize their result in two ways, by showing that a similar phenomenon occurs in both the general case of algebraic morphisms, and in the special case of word maps, where significantly stronger bounds are obtained. The (FRS) property has a deep connection to asymptotic point count over finite rings of the form Z/p^kZ. We utilize this connection to extract information on certain natural random walks induced by these word maps on compact p-adic groups, providing applications to p-adic probabilistic Waring-type problems. Based on a joint work with Yotam Hendel.Date: Thursday, October 31, 2019