joint with Geordie Williamson, We prove that (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients for ANY Coxeter system and ANY choice of a parabolic subgroup, thus generalizing to the parabolic setting the result The Hodge theory of Soergel bimodules by Elias and Williamson. We also prove a monotonicity conjecture of Brenti. The new techniques were used by Williamson and Lusztig to calculate many new elements of the p-canonical basis and thus make this conjecture. PDF The Anti-Spherical Category
I have been thinking about math education for more than fifteen years. An initiation rite for me was the following story. I was training three students for the math olympics in some school. They worked with me for one year without previous trainment. Then they went to the National Mathematical Olympiad in Chile and they obtained the three gold medals that were given that year. In that moment I was very moved and I realized that the method that I implemented (a method that has been evolving ever since) was interesting.
This paper is the first of a series of introductory papers on the fascinating world of Soergel bimodules. It is combinatorial in nature and should be accessible to a broad audience. PDF Gentle introduction to Soergel bimodules
Advances in Mathematics (2015) 772-807. I prove that Lusztig's conjecture reduces to a problem about the light leaves (as defined in my first paper). Using the result in Section 4.3 of this paper (that he discovered independently) Geordie Williamson disproved Lusztig's conjecture! The counterexamples grow exponentially in the Coxeter number. Here is Geordie's paper PDF Light leaves and Lusztig's conjecture
joint with Geordie Williamson, Comptes Rendus Mathematique Vol 355 (2017) Issue 8, 853-858. We prove that there are indecomposable Soergel bimodules (in type A) having negative degree endomorphisms. This is quite surprising and proves the existence of a non-perverse parity sheaf in type A.
Most of my work revolves around some beautiful and central objects in representation theory called Soergel bimodules. I have produced a basis of morphisms between such objects that are called Light Leaves. I am fascinated by them. On the one hand, it is because of them that one can actually compute in Soergel bimodule-land, and on the other, they have some very rich combinatorial structure that is slowly unearthing. My favorite subjects are diagrammatical category theory, categorical braid group actions, modular representations of finite or algebraic groups, Kazhdan-Lusztig theory, categorifications and knot theory. Very nice.
joint with Ben Elias ; with an appendix by Ben Webster, Trans. Amer. Math. Soc. 369 (2017), 3883-3910. We find the explicit numbers for which "Soergel's conjecture in positive characteristic for Universal Coxeter systems" fail, i.e. we find explicitly the projectors to the indecomposables. These are clearly the most simple Coxeter systems (after the dihedral groups) but even in this case the situation is quite subtle. It would be amazing (for modular representation theory) to find similar results as the ones in this paper, but for Weyl groups or affine Weyl groups. PDF Indecomposable Soergel bimodules for Universal Coxeter groups