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.
Joint with Gastón Burrull and David Plaza, submitted to IMRN. We prove the combinatorial invariance conjecture (by G. Lusztig and M. Dyer in the eighties) for the affine A2. This is the first infinite group with non-trivial Kazhdan-Lusztig polynomials where this fascinating conjecture is proved.
Joint with Leo Patimo and David Plaza. Accepted for publication in Advances in Mathematics. For any affine Weyl group, we introduce the pre-canonical bases, a set of bases of the spherical Hecke algebra that interpolates between the standard basis and the canonical basis. Thus we divide the hard problem of calculating Kazhdan-Lusztig polynomials (or q-analogues of weight multiplicities) into a finite number of much easier problems.
Joint with Geordie Williamson, Journal of Algebra 568 (2021) 181-192. When Soergel's conjecture is satisfied, we produce (finally!) the canonical light leaves, that do not depend on choices. This gives a new approach towards finding a combinatorial interpretation of Kazhdan-Lusztig polynomials.
Joint work with Gastón Burrull and Paolo Sentinelli. Advances in Mathematics 352 (2019) 246-264. In this paper we introduce the p-Jones Wenzl idempotent, a characteristic p analogue of the classical Jones-Wenzl idempotent. We hope this to be a building block for the p-canonical basis as sl_2 is a building block for the representation theory of semi-simple Lie algebras.
Joint work with David Plaza. Proc. Lond. Math. Soc. Vol. 121 (2020) Issue3, 656-701. We conjecture (and prove the "graded degree part") an equivalence between the type A affine Hecke category in positive characteristic and a certain blob category that we introduce as a quotient of KLR algebras. This conjecture has been proved recently in an amazing paper by Chris Bowman, Anton Cox, Amit Hazi!! It opens lots of questions...
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.
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.
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.