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

 

Featured Projects

 

Papers and preprints

  • The Anti-Spherical Category, joint with Geordie Williamson, We prove that parabolic Kazhdan-Lusztig polynomials have non-negative coefficients and a monotonicity conjecture of Brenti. The new techniques may be used to calculate many new elements of the p-canonical basis. 
  • Indecomposable Soergel bimodules for Universal Coxeter groups, joint with Ben Elias, with an appendix by Ben Webster, accepted for publication in Transactions of the AMS.  
  • 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.

  • Light leaves and Lusztig's conjectureAdvances in Mathematics  (2015), pp. 772-807
  • 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).  

  • Standard objects in 2-braid groups, joint with Geordie WilliamsonProc. London. Math. Soc. 109 (2014), no. 5, 1264-1280.   
  • 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

  • New bases of some Hecke algebras via Soergel bimodules Advances in Math. 228 (2011) 1043-1067.
  • 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.

  • Presentation of right-angled Soergel categories by generators and relations J. Pure Appl. Algebra 214 (2010), no. 12, 2265-2278.
  • 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". 

Organization of conferences

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.

 

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.

Students and postdocs


  • Gonzalo Jiménez, Ph.D stuedent January 2017 - Present

    Sebastián Cea, Master student September 2016 - Present

    David Plaza, Postdoc Sept 2013 - Jan 2016

    Paolo Sentinelli, Postdoc Dec 2015 - Present

    Gastón Burrul, Master student, June 2015- Present

    Pamela Pastén, Master student Sept 2014 - Present


Future talks and trips

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.

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