I am a lecturer at the University of New South Wales in the pure mathematics group.

Before this, I was a postdoctoral research associate and National Science Foundation postdoctoral fellow at the University of Sydney. I received my PhD in 2018 from the University of Utah under the supervision of Dragan Miličić.

I work in geometric and categorical representation theory. Some objects that I like to think about are Lie algebras, Lie groups, equivariant sheaves, D-modules, Soergel bimodules, and Kazhdan-Lusztig polynomials. Here is my CV.

News:

My little research group at UNSW is growing. Here's what we're up to.

I am helping to organise a workshop on diagrammatic categorification in October 2025 at ICERM.

In the first half of 2024, Alex Sherman and I ran a joint UNSW/USyd learning seminar on tensor categories and their modules. Notes from all the talks can be found here.

From August 2023 - January 2024, Dragan Miličić visited the University of Sydney and UNSW. With Geordie Williamson, he ran a course on D-modules and representation theory. Notes and recorded lectures can be found on the course website

In March 2023, I gave a public lecture on categorification at the Frontiers of Science Forum in Sydney. Before the event I chatted with Ian Woolf of Diffusion Radio about all things equality. You can listen to our conversation here.

I participated in the University of Sydney's Live from the Lab event in August 2021. I was paired with R&B soul artist Mi-Kaisha, who composed a beautiful piece of original music based off my research. To listen to her creation and hear about the process, check out the Live from the Lab podcast .

For much of 2019 and 2020, I typed notes for Geordie Williamson's 18 month lecture series on the Langlands correspondence and Bezrukavnikov's equivalence.

Come to the the Informal Friday Seminar!

In October 2019, I gave a public lecture as a part of Raising the Bar Sydney. My talk on Hidden Symmetries was recorded and made into a podcast.

An undergraduate project that I did with David Allen in 2011 recently appeared in Randall Munroe's new book How To. It's a wonderful book! If you get your hands on a copy, look for our flower-wheeled bicycle.

Whittaker modules for semisimple Lie algebras
I am interested in a category of modules over a complex semisimple Lie algebra which includes both Whittaker modules and highest weight modules. My PhD thesis gave a geometric algorithm for computing composition multiplicities of standard modules in this category. With Adam Brown, we established further properties of this category, including a classification of contravariant forms on standard modules and highest weight structures on blocks of the category. With Jens Eberhardt, we defined Jantzen filtrations of standard Whittaker modules and showed that they satisfy strong functoriality and semisimplicity properties. This proved a generalisation of the Jantzen conjectures for Whittaker modules. The algebraic tools I use to study Whittaker modules are similar to those used in the study of category O. With Sean Taylor, I wrote a book chapter on category O for the 2020 textbook Introduction to Soergel Bimodules, which was based on lectures given by Geordie Williamson and Ben Elias at a 2017 MSRI summer school on Soergel bimodules.

Two proofs of a Jantzen conjecture for Whittaker modules, (with J. Eberhardt). Selecta Mathematica. (To appear) (arXiv)

Contravariant pairings between standard Whittaker modules and Verma modules. (with A. Brown). Journal of Algebra, 609, 145-179, 2022. (arXiv)

Contravariant forms on Whittaker modules. (with A. Brown). Proceedings of the American Mathematical Society, 149(1), 37-52, 2020. (arXiv)

A Kazhdan-Lusztig algorithm for Whittaker modules. Algebras and Representation Theory, 24(1), 81-133, 2021. (arXiv)

A Kazhdan-Lusztig algorithm for Whittaker modules. PhD thesis, University of Utah (2018).

A lightning introduction to category O (with S. Taylor). Book chapter, Introduction to Soergel Bimodules. RSME Springer Series, Vol. 5, 2020.

D-modules and Beilinson-Bernstein localization
The main geometric tool in my work is the Beilinson-Bernstein localization theorem, which provides a link between representation theory of Lie algebras and D-modules on flag varieties. Viewing representation-theoretic concepts through the lens of D-modules can provide helpful geometric intuition and simplify arguments. Several of my papers pursue this philosophy. With Dragan Miličić, we reproved some classical theorems of Harish-Chandra about discrete series representations using simple D-module techniques. In 2020, I gave a detailed description of the D-modules corresponding to four families of representations, including principal series representations of SL(2,R) and Whittaker modules for sl(2,C). These computations provided a concrete geometric illustration of several classical families of representations. In 2024, I wrote a follow-up article with my honours student, Simon Bohun , which lifts these computations to base affine space and computes the Jantzen filtration on the corresponding D-modules.

On a geometric approach to the discrete series, (with D. Miličić). Proceedings of the Harish-Chandra Centenary Celebrations 2023, HRA Prayagraj. (To appear)

An example of the Jantzen filtration of a D-module, (with S. Bohun). Journal of the Australian Mathematical Society. 118(3), 265-296, 2025. (arXiv)

Four examples of Beilinson-Bernstein localization, Lie Groups, Number Theory, and Vertex Algebras, Contemporary Mathematics, vol. 768, Amer. Math. Soc., Providence, RI, 2021, pp. 65-85. (arXiv)

Soergel bimodules and real groups
Irreducible characters of real reductive algebraic groups can be computed using certain polynomials (Kazhdan-Lusztig-Vogan polynomials), which appear as entries in a change-of-basis matrix for the corresponding Lusztig-Vogan module of the Hecke algebra. With Scott Larson, we initiated a Soergel bimodule approach to the character theory of real groups by explaining how a piece of the Lusztig-Vogan module (the trivial block) admits a categorification in terms of bimodules over polynomial rings. Our bimodule category arises as the essential image of the hypercohomology functor applied to a category of constructible sheaves and admits a natural action of Soergel bimodules.

A categorification of the Lusztig-Vogan module. (with A. Larson). (arXiv)

Algebraic groups and OTI functors
The modular representation theory of algebraic groups is deep subject drawing on techniques from compact Lie groups, finite groups, and geometric representation theory, among other things. In 2023, I worked with a team of Sydney-based researchers to develop a new approach to the study of tensor products of modular representations. The approach introduces a functor (the so-called "OTI functor") which exhibits several useful properties. We conjecture that this functor provides an algebraic incarnation of hypercohomology under the Finkelberg-Mirkovic conjecture.

A remarkable functor on G-modules. (with J. Baine, T. Fell, A. Sherman, and G. Williamson). (arXiv)

Gelfand pairs
In 2016, I attended an AMS Mathematics Research Community (MRC) workshop on Lie groups, discretization, and Gelfand pairs. In this workshop, I started a project with a team of researchers that established a topological model for the space of spherical unitary representations of a nilpotent Gelfand pair in terms of coadjoint orbits in the dual Lie algebra. After the completion of this MRC project, I ran an REU (Research Experience for Undergraduates) out of the University of Utah on finite Gelfand pairs. With Faith Pearson and Dylan Soller, we studied families of group-subgroup pairs constructed from wreath products of symmetric groups, and established exactly when this construction yields a Gelfand pair.

Finite Gelfand pairs and cracking points of the symmetric groups. (with F. Pearson, and D. Soller). Rocky Mountain Journal of Mathematics, 50(5), 1807-1812, 2020. (arXiv)

An orbit model for the spectra of nilpotent Gelfand pairs. (with H. Friedlander, W. Grodzicki, W. Johnson, G. Ratcliff, B. Strasser, and B. Wessel). Transformation Groups, 25(3), 859-886, 2019. (arXiv)

Teaching
In 2025, I am on leave.
In Term 1 2024, I taught MATH 5735, Modules and Representation Theory.
In 2023, I was on leave.
In Term 2 2022, I taught MATH 1231, Mathematics 1B (linear algebra + probability).
In Term 1 2022, I taught MATH 5735, Modules and Representation Theory.

Student Supervision
My research group meets every Thursday from 9:30-11:00am in H13-4082. More information can be found here. Guests are welcome to join.

Current students:

Daniel Dunmore. Module categories over Soergel bimodules. PhD, UNSW (Primary supervisor)

Thomas Dumore. Quadratic fusion categories. PhD, UNSW (Secondary supervisor)

Victor Zhang. Diagrammatic Lusztig-Vogan categories. PhD, UNSW (Primary supervisor)

Past Students:

Tasman Fell. Singular Soergel bimodules. research masters, UNSW 2025 (Primary supervisor)

Zach Leong. Unitary dual of SL(2,R). honours, UNSW 2024

Simon Bohun. Beilinson-Bernstein Localization on base affine space. honours, UNSW 2023

Victor Zhang. Diagrammatic categories in representation theory. honours, UNSW 2023

Amy Bradford. Whittaker modules for affine Lie algebras. REU, University of Utah 2021

Dylan Soller. Cracking points of finite Gelfand pairs. REU, University of Utah 2019

Faith Pearson. Cracking points of finite Gelfand pairs. REU, University of Utah 2019