Special Year Learning Seminar

In these two talks, I will discuss the structure of certain varieties that depend functorially on the choice of a finite-dimensional vector space. Examples include the variety of d-way tensors of "slice rank" at most k and the variety of degree-d homogeneous forms of "border strength" at most k, where in both cases the underlying vector space varies.

A fundamental theorem says that such varieties satisfy the descending chain condition. The proof of this Noetherianity theorem reveals several further properties of such varieties: e.g., they are unirational up to a fixed finite-dimensional variety, and satisfy a version of Chevalley's theorem on constructible sets.

The ambient space for such a variety is given by a polynomial functor; in characteristic zero, these are the well-known Schur functors. I will start by discussing some basic properties of polynomial functors, and then discuss proofs of Noetherianity, unirationality, and related results.