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FIW–algebras generated in low degrees and convergence of point-counting

Banff International Research Station for Mathematical Innovation and Discovery

In this talk we will consider some families of varieties with actions of certain finite reflection groups -- varieties such as the hyperplane complements or complex flag manifolds associated to these groups. Beautiful results of Grothendieck–Lefschetz and Lehrer relate the topology of these complex varieties with point-counting over finite fields. Church, Ellenberg and Farb noticed that, in this context, representation stability in cohomology corresponds to asymptotic stability of various point counts over finite fields. The cohomology rings of the families considered have the structure of a graded FIW-module. We will discuss what this means, and how asymptotic stability is a direct consequence of the finite generation for FIW-algebras generated in degree at most one. We will also see that this is not necessarily the case when some of the generators are in degree two or higher. This is joint work with Jennifer Wilson.

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

video/mp4

Author affiliation: Universidad Nacional Autónoma de México

2016-10-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59397

Unreviewed

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