Professor Kirill Zaynullin (University of Ottawa)

From September 20 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 Memorial University and will be broadcast via Zoom. The schedule is tentatively:

• Tuesday, Sep. 21: 9-9:50 am and 3-3:50 pm (Atlantic time)
• Wednesday, Sep. 22: 9-9:50 am (Atlantic time)
• Thursday, Sep. 23: 9-9:50 am (Atlantic time)

For Zoom connection details and last minute schedule changes, see the event website.