## 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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## On reduced expressions for core double cosets

Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. We produce an algorithmic construction of a reduced expression for any double coset.

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## Singular light leaves

Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. For any Coxeter system we introduce the concept of singular light leaves (finally!!). Really fun stuff.

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## On the size of Bruhat intervals

Joint with Federico Castillo, Damian de la Fuente and David Plaza. For affine Weyl groups and elements associated to dominant coweights, we give a convex geometry formula for the size of the corresponding lower Bruhat intervals (this says in particular that it is a polynomial in several variables). Extensive computer calculations for affine Weyl groups have led us to believe that a similar formula exists for all lower Bruhat intervals. We also believe that the cardinality of all Bruhat intervals is given by some family of quasi-polynomials.

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## Subexpressions and the Bruhat order for double cosets

Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. This is the second paper on the development of a singular Coxeter theory. Here we give a singular analogue of the usual subexpression version of the Bruhat order.

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## Demazure operators for double cosets

Joint with Ben Elias, Hankyung Ko and Leonardo Patimo. This is the first of a series of papers intended to advance in the development of a singular (i.e. double cosets) theory of Coxeter groups, Hecke algebras, actions of groups on polynomial rings and the Hecke category. Two milestones of this long-term project would be to produce singular light leaves and singular Soergel calculus. Here, we introduce a Demazure operator for any double coset. We prove several results about them, but the crucial thing for us is that they give a criterion for ensuring the proper behavior of singular Soergel bimodules.

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## On the affine Hecke category for SL(3)

Joint with Leonardo Patimo, Selecta Mathematica 29 (2023), no. 4, Paper No. 64. We find a surprisingly beautiful basis of the Hom spaces between indecomposable Soergel bimodules for SL(3) (something that we call indecomposable light leaves.

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## Introsurvey of Representation Theory

Accepted for publication in Journal of the Indian Institute of Science. It is an introduction/survey of representation theory with lots of humor, open questions, and extremely smart insights :)

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## Combinatorial invariance conjecture for affine A2

Joint with Gastón Burrull and David Plaza, International Mathematics Research Notices 10 (2023) 8903–8933.  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.

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## Pre-canonical bases on affine Hecke algebras

Joint with Leonardo Patimo and David Plaza, Advances in Mathematics  399 (2022). 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.

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## The Anti-Spherical Category

joint with Geordie Williamson, Advances in Mathematics  405 (2022). 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.

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## Kazhdan-Lusztig polynomials and subexpressions

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.

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## Blob algebra approach to modular representation theory

Joint work with David Plaza. Proceedings of the London Mathematical Society 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...

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## p-Jones Wenzl idempotent

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.

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## Gentle introduction to Soergel bimodules I: The basics

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.

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## A non-perverse Soergel bimodule in type A

joint with Geordie WilliamsonComptes 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.

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## Our article with Jorge Soto-Andrade

Here is the first (and only until now) research article I did about these ideas, with Jorge Soto-Andrade On the…

## 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.