In analysis, the convolution of two functions results in a smoother, better behaved function. It is interesting to ask whether an analogue of this phenomenon exists in the setting of algebraic geometry.
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 $X \times Y$ to $G$ by first applying each morphism and then multiplying using the group structure of $G$.
In this talk, we present some properties of this convolution operation, as well as a recent result which states that after finitely many self convolutions every dominant morphism $f:X \to G$ from a smooth, absolutely irreducible variety $X$ to an algebraic group $G$ becomes flat with reduced fibers of rational singularities (this property is abbreviated FRS).
The FRS property is of particular interest since by works of Aizenbud and Avni, FRS morphisms are characterized by having fibers whose point count over the finite rings $\mathbb{Z}/p^k\mathbb{Z}$ is well-behaved. This leads to applications in probability, group theory, representation growth and more.
We will discuss some of these applications, and if time permits, the main ideas of the proof which utilize model-theoretic methods.
Joint work with Itay Glazer.