About me
I studied my undergrad in the University of Chile and in the École Normale Supérieure where I also started my Ph.D under the supervision of Raphaël Rouquier. I finished my Ph.D in Paris VII. Afterwards I did a two year Von Humboldt postdoc under the supervision of Wolfgang Soergel. My CV is available here. Here a video about my work, appeared in CNN Chile
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.
Using a special case of 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 his paper
I prove that Lusztig's conjecture reduces to a problem about the light leaves (as defined in my first paper).
For any Coxeter system, we establish the existence of analogues of standard and costandard objects in 2-braid groups, thus proving a conjecture that Rouquier stated in the ICM 2006. This result was a key step for the proof of Kazhdan-Lusztig conjectures
This is a first attempt to find explicitely Soergel indecomposable bimodules for extra-large Coxeter systems. This is very linked with my "Forking path conjecture" (see the paper "Gentle Introduction to Soergel bimodules" above), an extremely strange phenomenon that I would love to understand better.
I find a presentation of Soergel category (as a tensor category) by generators and relations in the right-angled Coxeter group case. This problem was solved in complete generality in the beautiful paper Soergel calculus. In simple words this could be summarized as "how to draw Soergel bimodules".
I prove that in Soergel's conjecture it is equivalent to use the "easy" geometric representation or the "difficult" reflection faithful representation used before.
I introduce the light leaves basis, a basis of the Hom space between Soergel bimodules. They have been useful in the disproof of Lusztig and James conjectures, in the algebraic proof of KL conjecture, in the proof of the positivity of the coefficients of KL polynomials (and its parabolic version), in the algebraic proof of Jantzen conjecture, and in many other contexts. It has also been a key ingredient to the new approach to modular representation theory by Williamson and Riche.
Chapter 1 is a essentially a version of this paper (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.
- 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.
Invited talk in the Conference "La resolución de Problemas, una competencia transversal", 26th May 2017, Santiago, Chile.
Invited stay in Sydney University, invited by Geordie Williamson, 5th July-25th August.
Invited talk in "Math Congress of the Americas" in the session "Representations of Lie algebras", 28th July 2017.
Plenary talk in "Representation Theory Day in Québec City", 29th July 2017.
Featured Projects
Coorganizer (with Aaron Lauda and Joshua Sussan) of the session "Symmetry in algebra, topology, and physics" in the Math Congress of the Americas
"Quinquagenary, Faculty of Sciences, University of Chile", 9-11 December, 2015, Santiago, Chile. (I invited ten Nobel prize recipients to this conference).
Primer Congreso Internacional Aproximaciones Experimentales a la Interacción Social, 14-16 January 2015, Valparaíso, Chile.
2013 - 2015 Weekly colloquium, every week, Santiago, Chile
Orderable groups, 1-5 September 2014, Cajón del Maipo, Chile.