Abstract for the talk on 26.01.2021 (17:00 h)Nonlinear Algebra Seminar Online (NASO)
Jan Draisma (University of Bern)
Algebra over FI
See the video of this talk.
FI is the category of finite sets with injections. An FI something F is a functor from FI to the category of somethings. In particular, for S in FI, the symmetric group Sym(S) acts on F(S) by something automorphisms of F(S). FI-somethings arise in several areas of mathematics; for instance, FI-modules arise as cohomology groups of configuration spaces of n labelled points on a given manifold (with n varying), and FI-algebras arise as coordinate rings of certain algebraic-statistical models, where the number of observed random variables varies. This talk aims to be a gentle introduction into algebra over FI; no prior exposure to FI-maths is required.
Sometimes, FI-somethings inherit good properties of the category of somethings (and indeed the same applies when FI is replaced by other suitable base categories). I will present several, by now, classical examples of this phenomenon, primarily due to Church, Ellenberg, Farb, Daniel Cohen, Aschenbrenner, Hillar, Sullivant, Sam, and Snowden. For instance, a finitely generated FI-module M over a Noetherian ring is Noetherian, and if the ring is a field, then the dimension of M(S) is eventually a polynomial in |S|.
After this overview, I will zoom in on new joint work with Rob Eggermont and Azhar Farooq that says that if X(S) < K^c x S is avariety of c x S-matrices and for every injection S -> T the natural map K^c x T -> K^c x S maps X(T) into X(S), then the number of Sym(S) orbits on the set of irreducible components of X(S) is a *quasi*polynomial in |S| for |S| sufficiently large.