“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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Light leaves and Lusztig’s conjecture

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  

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Standard objects in 2-braid groups

Joint with Geordie WilliamsonProc. London. Math. Soc. 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

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New bases of some Hecke algebras via Soergel bimodules 

Advances in Math. 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. If my "Forking path conjecture" (see the paper "Gentle Introduction to Soergel bimodules" above) is correct, the same kind of bases would exist for the symmetric group. I believe that they have a deep meaning related to the Hecke category and the p-canonical basis.

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Presentation of right-angled Soergel categories by generators and relations

J. Pure Appl. 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.

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My Ph.D thesis

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.

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