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FI-modules and counting points over finite fields

Banff International Research Station for Mathematical Innovation and Discovery

It is well-known by now that homological stability for a sequence of moduli spaces \(X_1, X_2, X_3\), over a finite field \(F_q\) sometimes translates into the existence of a limit for the point-count \[ q^{-dim X_n} |X_n(F_q)|. \] Sometimes you can tell a similar story about sequences of moduli spaces whose cohomology is representation (or: naturally forms a finitely generated module for some FI-like category.) This provides, for example, a geometric way of thinking about the distribution of irreducible factors of random polynomials and of irreducible factors of random maximal tori in algebraic groups over \(F_p\). I will also talk about work of Vlad Matei on the function-field Landau problem — how often is a random polynomial a sum of two squares? — and explain what this has to do with the category FI_B introduced by J. Wilson.

Mathematics; Commutative rings and algebras; Category theory; homological algebra; Commutative algebra

video/mp4

Author affiliation: University of Wisconsin

2016-10-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59348

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/