Gaston Burrull's personal website 

Research Interests

• My research interests are in geometric representation theory. My current research topic is the combinatorics of Schubert varieties and Kazhdan-Lusztig polynomials, I am particularly interested in Lusztig-Dyer's combinatorial invariance conjecture. I am interested in visualizations of difficult mathematical concepts. I use coding to produce visualizations and gain understanding.

Publications and preprints

Publications

1. Combinatorial invariance conjecture for A˜2
   Joint with Nicolas Libedinsky and David Plaza, International Mathematics Research Notices. 10 (2023), 8903–8933. https://doi.org/10.1093/imrn/rnac105
   In this paper, we proved in type A˜2 an old conjecture stated independently by G. Lusztig and M. Dyer. This conjecture says that if the intervals [x,y] and [x',y'] are isomorphic as Bruhat posets, then the corresponding Kazhdan-Lusztig polynomials P_{x,y}(q) and P_{x',y'}(q) should be equal. Here are the slides of my 3-minute talk about this work at the AMSI Winter School 2022 - New Directions in Representation Theory.

2. p-Jones-Wenzl idempotents
   Joint with Nicolas Libedinsky and Paolo Sentinelli, Adv. Math. 352 (2019), 246–264. https://doi.org/10.1016/j.aim.2019.06.005
   In this paper, we find a characteristic p-analog of the classical recursion formula for the Jones-Wenzl idempotents. Here is my illustrative poster about this work which I presented at the New Connections in Representation Theory conference.

Preprints

1. Strongly dominant weight polytopes are cubes
   Joint with Hongsheng Hu and Tao Gui, preprint (2023).
   https://arxiv.org/abs/2311.16022

2. Asymptotic Log-concavity of Dominant Lower Bruhat Intervals via the Brunn–Minkowski Inequality
   Joint with Hongsheng Hu and Tao Gui, preprint (2023). Accepted as an extended abstract in Formal Power Series and Algebraic Combinatorics 2024.
   https://arxiv.org/abs/2311.17980

Notes

• On the combinatorics of parabolic Schubert varieties. This is my Ph.D. thesis, officially submitted in January 2023. Chapter 5 includes the examination reports from the thesis referees Jacob Matherne and Leonardo Patimo together with my answers. After the official submission, some minor mistakes were found, fixed, and listed in Chapter 6. Any comments are welcome. Last version: 13 July 2023. 

• Combinatorial invariance conjecture for A˜2. These are the slides of the talk I gave in 2022 at the AMSI Winter School 2022 - New Directions in Representation Theory.

• My Ph.D. final presentation. June 2022.

• Top-heaviness of Bruhat intervals. These are the slides of the talk I gave in 2022 at the Algebra seminar.

• Unimodality of Bruhat intervals. These are the slides of the talk I gave in 2021 at the 65th Annual Meeting of the Australian Mathematical Society.

• Hard Lefschetz and the shape of the Bruhat intervals. These are the notes of the second talk I gave in 2021 in the Informal Friday Seminar organised by Bregje Pauwels. (21 pages, iPad notes)

• Introduction to Schubert calculus and flag varieties. These are the notes of a talk I gave in 2021 in the Informal Friday Seminar organised by Bregje Pauwels. (18 pages, iPad notes)

• Classical motivation for Intersection Homology. These are the notes of a talk I gave in 2021 in the Student Algebra Seminar organised by Joshua Ciappara and Bregje Pauwels. (8 pages, iPad notes)

• T-structures, hearts, and the recollement situation. These notes are based on a talk I gave in 2021 in the Perverse Sheaves Seminar organised by Michael Zhao. (8 pages, iPad notes)

• What is modular representation theory?. These are some notes I took in 2020 in the What is ... ? Seminar of a talk given by Joshua Ciappara. These are a nice introduction to the representation theory subject. (7 pages, iPad notes)

• Introduction to derived categories of coherent sheaves. Notes from a course given by Tom Bridgeland in Sydney Mathematical Research Institute, 2020. Notes by Linyuan Liu and me. (15 pages, LaTeX notes)

• The Universal enveloping algebra II. These are the notes of a talk I gave in 2020 in the Student Algebra Seminar organised by Joel Gibson and Joshua Ciappara. (7 pages, iPad notes)

• The Universal enveloping algebra I. These are the notes of a talk I gave in 2020 in the Student Algebra Seminar organised by Joel Gibson and Joshua Ciappara. (5 pages, iPad notes)

• Chamber systems and the Coxeter complex. These are the notes of a talk I gave in 2020 in the Student Algebra Seminar organised by Joel Gibson and Joshua Ciappara. (7 pages, iPad notes)

• The Bruhat graph of a Coxeter group. These are the notes of a talk I gave in 2019 in the Informal Friday Seminar organised by Anna Romanov. (4 pages, notebook notes)

• The Bernstein presentation of the Hecke algebra. These notes are based on a talk I gave in 2019 in the Informal Friday Seminar organised by Anna Romanov. (6 pages, LaTeX notes)

• A beautiful proof that the Cantor set is not countable. (2 pages, iPad notes)

Posters

• My poster explaining the "p-Jones-Wenzl idempotents" of the Temperley-Lieb algebra. I presented this poster at the New Connections in Representation Theory conference held at Mantra Hotel Conference Centre in February 2020.

Notebook notes

Some of these are notebook notes I'd be happy to convert to TeX. I put them here as a backup.

• Schubert calculus, Deodhar varieties, reductive groups. (38 pages)

• Commutative algebra, affine schemes, Schubert cells, Grassmannians, tangent space, projective schemes, weights and vector bundles. (48 pages)

• Notes on parabolic R-polynomials, combinatorics of Coxeter groups and buildings. (27 pages)

Related/Useful links

• David Kazhdan's personal website. My postdoctoral supervisor's website.

• Geordie Williamson's personal website. My Ph.D. supervisor's website.

• Nicolas Libedinsky's personal website. My master and Ph.D. supervisor's website.

• Anthony Henderson's personal website. My auxiliary Ph.D. supervisor's website.

• Oded Yacobi's personal website. My auxiliary Ph.D. supervisor's website.

• Igor Balla's personal website. My friend Igor, a postdoctoral researcher at the Einstein Institute of Mathematics of the Hebrew University of Jerusalem, hosted by Eran Nevo and Alex Samorodonitsky.

• Paolo Sentinelli's personal website. My friend Paolo, a postdoc at Politecnico di Milano.

• Kane Townsend's personal website. My friend Kane, a postdoc at UTS working under Murray Elder. 

• Leonardo Maltoni's personal website. My friend Leonardo, a former student of Daniel Juteau and Geordie Williamson.

• Leonardo Patimo's personal website. He was a Ph.D. student of Geordie Williamson. Currently, a Research Assistant (of Prof. Wolfgang Soergel) at the University of Freiburg.

• Joshua Ciappara's personal website. He was a Ph.D. student of Geordie Williamson.

• Advice on writing. Given by Professor Terence Tao.

• Sage Reference Manual.

• Inkscape. A free tool for drawing vector graphics.

• The Stacks project. One of the best references for algebraic geometry.

• How to give a good colloquium. By Professor John E. McCarthy.

• n-lab.

• n-category cafe.

• If you are a non-native English speaker the Cambridge Dictionary and this video about vowel sounds might be helpful for you.

Other files

• My master's thesis. Supervised by Nicolas Libedinsky.