Advances in Mathematics (2015) 772-807. I introduce the double leaves basis and with it I prove that Lusztig's conjecture reduces to a problem about the light leaves. Using the result in Section 4.3 of this paper Geordie Williamson disproved Lusztig's conjecture! The counterexamples grow exponentially in the Coxeter number. Here is Geordie's paper
Orderable groups, 1-5 September 2014, Cajón del Maipo, Chile Minicourses: 1) Ordered groups and topology. Adam Clay (U. Québec / Montréal). 2) Orderable group actions and the deep-fall property. Igor Mineyev (Illinois…
Joint with Geordie Williamson, Proceedings of the London Mathematical Society 109 (2014), no. 5, 1264-1280. We introduce the concept of Δ-exact complexes for any Coxeter system. With that, we establish the existence of analogues of standard and costandard objects in 2-braid groups, thus proving the conjecture that Rouquier stated in the ICM 2006. This result was a key step for the proof by Elias and Williamson of Kazhdan-Lusztig conjectures
2013 - 2015 Weekly colloquium, every week, Santiago, Chile 2015 9 Diciembre Cédric Villani (IHP) "The living art of mathematics". 2 Diciembre Eduardo Friedman (U. de Chile, Ciencias) "Co-volumen de subgrupos de las unidades de un…
Advances in Mathematics 228 (2011) 1043-1067. I introduce a new set of bases for Hecke algebras related to extra-large Coxeter groups, coming from the theory of Soergel bimodules. I believe that they have a deep meaning related to the Hecke category and the p-canonical basis. This is related to my "Forking Path conjecture" that is false as stated, as proved by Gonzalo Jimenez. Nonetheless, in all generality, there seems to be an extremely strange phenomenon that I would love to understand better.
Journal of Pure and Applied Algebra 214 (2010) no. 12, 2265-2278. This was the first time that a "presentation of Soergel bimodules by generators and relations" was attempted. This revolutionary idea (explained to me by Rouquier) was the key of all the impressive subsequent development of the theory.
Chapter 1 is essentially a version of the paper that one could call "Soergel bimodules explained by Soergel" with explanations of the obscure points. Sections 2.4 and 2.5 are original and are not included in any other paper. I give a different (and easier) proof of the fact that Rouquier complexes satisfy the braid relations.
Journal of Algebra, 320 (2008) 2695-2705. I prove that in Soergel's conjecture it is equivalent to use the "easy" geometric representation or the "difficult" reflection faithful representation used before.
Journal of Algebra 320 (2008) 2675-2694. I introduce the light leaves basis.