Estuvimos a punto de ejecutar un trabajo perfecto, Nathalie en una casa de piedra de Provenza. Dirás ahora que todo…
joint with Geordie Williamson, submitted to Advances in Mathematics. 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 central result of 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 the Billiards conjecture. Along the way, we introduce the anti-spherical light leaves.
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.
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.
Sao Paulo Journal of Mathematical Sciences, 13(2) (2019), 499-538. 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. We introduce the Forking path conjecture.