Andrei Okounkov: 14-16 November 2016

As part of the Coxeter Lecture Series, Andrei Okounkov will give a series of three lectures related to the thematic program.

Counting in representation theory and geometry

by Andrei Okounkov (Columbia)
Monday, 14 November from 15:30 to 16:30 in Room 230 at the Fields Institute

The character of a representation enumerates the eigenvalues of the operators by which a group G acts. A geometric generalization of a linear representation is an action of a group on a vector bundle V over some base X. Given an element g of G, we can ask how it acts on fibers over points fixed by g; we can also consider the action of g on global sections and, more generally, higher cohomology groups of V. Localization theorem of Atiyah and Bott is the first is a long series of important results connecting the two, and it will be the starting point of our discussion. My next goal will be to explain why is it interesting in enumerative geometry and which form does it take there.

Localization and rigidity in enumerative geometry

by Andrei Okounkov (Columbia)
Tuesday, 15 November from 15:30 to 16:30 in Room 230 at the Fields Institute

Classical ideas of Atiyah and Hirzebruch, further developed by Krichever, Witten, Bott-Taubes, and many others, show that sometimes the representations computed in Lecture 1 are trivial. I will explain why this triviality is nontrivial and, in fact, quite exciting for enumerative geometers, and how it is put to practical use in several situations.

Geometry and combinatorics of stable envelopes

by Andrei Okounkov (Columbia)
Wednesday, 16 November from 15:30 to 16:30 in Room 230 at the Fields Institute

To make full use of the techniques explained in Lecture 2, one needs a supply of K-theory classes satisfying nearly conflicting bounds on their supports and weights at fixed points. Such classes, called stable envelopes, play an important role in geometric representation theory where they give a geometric constructions of tensor products and braidings. The emphasis in this lecture will be on combinatorial aspects of stable envelopes.

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