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.
2013 - 2015 Weekly colloquium, every week, Santiago, Chile
Orderable groups, 1-5 September 2014, Cajón del Maipo, Chile.
Primer Congreso Internacional Aproximaciones Experimentales a la Interacción Social, 14-16 January 2015, Valparaíso, Chile.
"Quinquagenary, Faculty of Sciences, University of Chile", 9-11 December, 2015, Santiago, Chile. (I invited ten Nobel prize recipients to this conference).
- Soergel's category of bimodules, mostly in their relation with representations of algebraic groups, with the Kazhdan-Lusztig (KL) polynomials and to 2-braid groups. I believe that the light leaves basis introduced in my first paper is an important tool that should be studied in detail. This has been shown on the one hand by the proof given in 2012 by B. Elias and G. Williamson's of the KL positivity conjectures and their algebraic proof of KL conjecture, and on the other hand, by my paper "Light leaves and Lusztig's conjecture" that reduces Lusztig's conjecture to a light leaves problem. G. Williamson's disproof of Lusztig's conjecture uses this approach. It seems that the p-canonical basis, controls de representations of algebraic groups and this basis is controlled by the light leaves.
- With Geordie Williamson we have been thinking about 2-braid groups as defined by R. Rouquier. In particular, we would like to understand the interaction between its algebraic and topologic properties, how the Garside structure in braid groups can be "seen" in the 2-braid groups context and Rouquier's conjecture saying that 2-braid groups categorify braid groups for every Coxeter system
- With Ben Elias we have been working in Soergel's conjecture in positive characteristic for Universal Coxeter systems and for affine Weyl groups.
- With Luis Arenas-Carmona we are working in different topics around light leaves basis and p-canonical bases.
- Director of RIA a multidisciplinary project aimed to understand the collaboration phenomenon and the "group intelligence". This project is composed by linguists, economists, a primatologist, educational psychologists and some other researchers. We already won a two year consolidation funding Domeyko II.
The last week of June 2013 I will stay in Paris, France.
All the month of July 2013 I will stay in the Max Planck Institute in Bonn, Germany.
Mini-course, 26-30 August 2013 invited by Vyacheslav Futorny in Sao Paulo University, Brazil.
Invited talk in Representation theory days in Patagonia, 20-24 January 2014, Punta Arenas, Chile.
Invited talk in Catholic University seminar, 3 June 2014, Santiago, Chile.
Invited talk in Talca University seminar, 26 June 2014, Talca, Chile.
Visit, 5-10 July 2014, Paris, France.
Visit, 10-20 July 2014 to Max Planck in Bonn, Germany.
Mini-course, 4-8 August 2014 VII Argentinian national congress of Algebra, Sierras de Córdoba, Argentina.
Mini-course and two invited talks in XX latinoamerican colloquium of algebra, 8-12 December 2014, Lima, Perú.
Visit 1-8 February, Paris, France.
Invited talk in Quantum 2015, 2-6 March 2015, Córdoba, Argentina.
Invited talk in Enveloping Algebras and Geometric Representation Theory, 10-16 May 2015, Oberwolfach, Germany.