Joel Gibson - Research

I am currently a postdoctoral researcher at the Sydney Mathematical Research Institute, working with Geordie Williamson. My interests are in algebraic and geometric aspects of representation theory.

Mathematics software

I list most mathematics software I’ve published on my home page. You might be most interested in my Kronecker coefficient calculator or my visualisation software Lievis, which produces pictures like these and many more:

A simple character of $B_2$ over $\mathbb{F}_5$

Publications

Calculating the $p$ -canonical basis of Hecke algebras, Preprint (accepted into Transformation Groups) with Thorge Jensen and Geordie Williamson. The Magma package ASLoc is available.
Appendix of Categorical diagonalization and $p$ -cells, Preprint with Ben Elias and Thorge Jensen.
A Demazure character formula for the product monomial crystal, Algebraic Combinatorics, Volume 4 (2021) no. 2, pp. 301-327. DOI 10.5802/alco.156.
The Hecke Algebra and Kazhdan–Lusztig Polynomials (with Alexander Kerschl). Book chapter, Introduction to Soergel Bimodules, RSME Springer Series, Vol. 5, 2020. DOI 10.1007/978-3-030-48826-0.
Why Non-blocking Operations Should be Selfish. In: Moses Y. (eds) Distributed Computing. DISC 2015. Lecture Notes in Computer Science, vol 9363. Springer, Berlin, Heidelberg. DOI 10.1007/978-3-662-48653-5_14, pdf link

Other writing

In 2020 I completed my PhD thesis A Demazure character formula for the product monomial crystal under the supervision of Dr Oded Yacobi.
In 2015 I completed my honours thesis Contention in concurrent data structures under the supervision of Dr Vincent Gramoli.

Talks

I gave a talk on the product monomial crystal at AustMS 2018, held at the University of Adelaide, which was one of the two winners of the Bernhard Neumann prize for most outstanding student talk. Here are both the slides and annotated slides for the talk. (To view the annotated slides, open your PDF reader in “two pages per screen” mode).

Posters

A poster explaining a problem in combinatorial commutative algebra to do with the embedding of the affine Grassmannian of $\operatorname{SL}_n$ into the infinite (or Sato) Grassmannian. This was presented at the New Connections in Representation Theory conference held in Mooloolaba, Queensland in 2020. I experimented with making this poster in HTML rather than LaTeX and it turned out rather well – feel free to investigate the (a little messy) source code. HTML, PDF.
A poster on a conjecture (now a theorem) on the representation afforded by a graded Nakajima quiver variety. This was presented at the Future Directions in Representation Theory conference held at the University of Sydney in 2017. PDF

Conferences

You might have met me at the following conferences or workshops:

The 2022 AMSI Winter School - Brisbane, Queensland, June 2022.
New connections in representation theory - Mooloolaba, Queensland, February 2020.
Flags, galleries, and reflection Groups - The University of Sydney, August 2019.
Triangulated categories in geometry and representation theory - The University of Sydney, June 2019.
Geometric and categorical representation theory - MATRIX, December 2018.
AustMS 2018 - University of Adelaide, December 2018.
Categorification and higher representation theory - Institut Mittag-Leffler, Stockholm, July 2018.
Symplectic duality (the Abelian case) - WARTHOG (University of Oregon), August 2017.
Future directions in representation theory - University of Sydney, December 2017.
Soergel bimodules - MSRI summer school, July 2017.
W-algebras - University of Melbourne, December 2016.