## Gonzalo Jiménez

Ph.D student January 2017 – Present

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# For Mathematicians

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Sebastián Cea

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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).

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

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Light leaves and Lusztig’s conjecture

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Orderable groups, 1-5 September 2014, Cajón del Maipo, Chile

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Standard objects in 2-braid groups

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2013 – 2015 Weekly colloquium, every week, Santiago, Chile

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New bases of some Hecke algebras via Soergel bimodules

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Presentation of right-angled Soergel categories by generators and relations

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My Ph.D thesis

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Équivalences entre conjectures de Soergel

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Sur la catégorie des bimodules de Soergel

Master student September 2016 – Present

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September 1, 2016

"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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December 9, 2015

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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January 14, 2015

**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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January 1, 2015

Orderable groups, 1-5 September 2014, Cajón del Maipo, Chile Minicourses: 1) Ordered groups and topology. Adam Clay (U. Québec / Montréal).…

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September 1, 2014

Joint with Geordie Williamson, **Proc. 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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January 1, 2014

2013 - 2015 Weekly colloquium, every week, Santiago, Chile 2015 9 Diciembre Cédric Villani (IHP) "The living art of mathematics". 2 Diciembre Eduardo Friedman (U. de…

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January 20, 2013

**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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June 29, 2011

**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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January 27, 2009

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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November 17, 2008

**Journal of Algebra**, 320 (2008) 2695-2705.
I prove that in Soergel's conjecture it is equivalent to use the "easy" geometric representation or the "difficult" reflection faithful representation used before.

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July 30, 2008

**Journal of Algebra** 320 (2008) 2675-2694.
I introduce the **light leaves basis. **

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July 14, 2008