Convolutions of algebraic morphisms and applications

In analysis, the convolution of two functions results in a smoother, better behaved function. A natural question is then whether this phenomenon has an analogue in the setting of algebraic geometry.

Let f,g be two morphisms from algebraic varieties X,Y to an algebraic group G. We define their convolution to be a morphism f*g from X x Y to G by first applying each morphism to its respective coordinate and then multiplying using the group structure of G.

In this talk, we will present some properties of this convolution operation, as well as a recent result which states that, under mild conditions, after sufficiently many self convolutions every morphism f:X->G becomes flat, with reduced fibers of rational singularities (abbrevieted FRS). This gives a possible answer to the question above.

In addition, the FRS property is of particular interest since, by works of Aizenbud and Avni and of Mustata, it has close ties to the asymptotic point count of the fibers of f over Z/p^kZ. This connection allows us to draw interesting conclusions on algebraic families of random walks on finite groups.

Joint work with Itay Glazer.