Highlighted Papers

The Anti-Spherical Category

joint with Geordie Williamson, 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 result 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 this conjecture.

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My Education Manifesto

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.

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Blob algebra approach to modular representation theory

Joint work with David Plaza. We provide a conjecture (and the proof of the "graded degree part") that links the Hecke category in positive characteristic and a certain "blob category" that we introduce as a quotient of KLR algebras. A part of the latter category can be understood in physical terms when the characteristic is zero. If there was a physical interpretation in the positive characteristic case, it would be majestic, because one could have physical intuition in one of the most difficult problems in representation theory (understanding the p-canonical basis).

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Research interests

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.

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Coorganizer (with Aaron Lauda and Joshua Sussan) of the session “Symmetry in algebra, topology, and physics” in the Math Congress of the Americas

Coorganizer (with Aaron Lauda and Joshua Sussan) of the session "Symmetry in algebra, topology, and physics" in the Math Congress of the…

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Indecomposable Soergel bimodules for Universal Coxeter groups

joint with Ben Elias ; with an appendix by Ben Webster,  Trans. Amer. Math. Soc. 369 (2017), 3883-3910. 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.

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“Quinquagenary, Faculty of Sciences, University of Chile”, 9-11 December, 2015, Santiago, Chile. (I invited ten Nobel prize recipients to this conference).

"Quinquagenary, Faculty of Sciences, University of Chile", 9-11 December, 2015, Santiago, Chile. (I invited ten Nobel prize recipients to this conference).

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Primer Congreso Internacional Aproximaciones Experimentales a la Interacción Social, 14-16 January 2015, Valparaíso, Chile.

Primer Congreso Internacional Aproximaciones Experimentales a la Interacción Social, 14-16 January 2015, Valparaíso, Chile. Congreso Internacional, Aproximaciones Experimentales a la Interacción…

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