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Atlantic Association for Research in the Mathematical Sciences
Atlantic Algebra Centre 

Introduction to Schubert calculus
via (nil-)Hecke algebras

Mini course by

Professor Kirill Zaynullin

University of Ottawa

September 21 - 23, 2021

From September 21 to September 23, 2021, Professor Kirill Zaynullin from the University of Ottawa will give an introductory mini- course on nil-Hecke algebras and their applications in cohomology.

The mini-course will consist of four lectures and will give a self-contained exposition on the use of the techniques of nil-Hecke algebras in the equivariant Schubert calculus for cohomology of flag varieties.

The first part will discuss root datum and Coxeter groups (Lectures 1-2): definition of a root datum, simple roots, fundamental weights and the Cartan matrix, the Dynkin diagram, the Weyl group, geometric realization, finite real root systems, coefficient ring of a root system, non-crystallographic root datum.

The second part will introduce nil-Hecke rings and twisted group algebras (Lectures 2-3): definition of nil-Coxeter and nil-Hecke rings, twisted group algebras and their localizations, coproducts, Hecke and Weyl actions, characteristic and the Borel maps. 

The third part (Lectures 3-4) will relate nil-Hecke rings and the Schubert calculus techniques: push-pull elements and divided- difference operators, the coproduct and the actions, faithful representation, the augmented coproduct and the formula for the coproduct, the dual of the nil-Hecke ring and equivariant cohomology. 

The lectures will take place at the St. John's campus of ÐÓ°É´«Ã½ University and will be broadcast via Zoom. The schedule is as follows in Newfoundland Time:

Tuesday, September 21, 9:30-10:20 am () and 3:30-4:20 pm (Lecture 2 notes);

Wednesday, September 22, 9:30-10:20 am ();

Thursday, September 23, 9:30-10:20 am ().

The room is HH-3015 or join via Zoom.