Kazhdan-Lusztig polynomials are fascinating objects in representation theory. They have interesting connections to the geometry of Schubert varieties and to combinatorics. This course will provide an introduction to Coxeter groups, their Hecke algebras, its Kazhdan-Lusztig basis, and Kazhdan-Lusztig polynomials. The second half of the course will focus on the combinatorial invariance conjecture, which is a fascinating conjecture concerning Bruhat graphs (certain directed graphs built out of Coxeter groups), and Kazhdan-Lusztig polynomials.
Some mathematical maturity will be assumed. If you want to get started, the book “Introduction to Soergel bimodules” by Elias, Makisumi, Thiel and Williamson is a good place to start.
Geordie Williamson works in Representation Theory, the mathematical theory of linear symmetry. He has made several fundamental contributions to the field including his proof (with Ben Elias) of the Kazhdan-Lusztig positivity conjecture, his algebraic proof of the Jantzen conjectures, and his discovery of counter-examples to the expected bounds in the Lusztig conjecture in modular representation theory.
For his work he has been awarded the Chevalley Prize of the American Mathematical Society, the European Mathematics Society Prize, the Clay Research Award and the New Horizons Prize in Mathematics (with Ben Elias). In 2018 he addressed the International Congress of Mathematicians as a plenary speaker. More recently, he received the medal of the Australian Mathematical Society and the Christopher Heyde Medal of the Australian Academy of Science.
Geordie is currently Professor of Mathematics at the University of Sydney. Prior to coming to Sydney he spent five years as an Advanced Researcher at the Max Planck Institute in Bonn. In 2020 he led a special year on Representation Theory at the Institute for Advanced Study in Princeton.