### About Me

I'm an assistant professor at the University of Southern California. I work on problems in representation theory, geometry, and combinatorics that are informed by theoretical physics, and am supported by NSF CAREER grant DMS-2143922.

### Research

Papers and preprints:

- Differential operators on the base affine space of SL_n and quantized Coulomb branches, with Tom Gannon. (arXiv)
- Canonical bases for Coulomb branches of 4d N=2 gauge theories, with Sabin Cautis. (arXiv)
- Tamely presented morphisms and coherent pullback, with Sabin Cautis. (arXiv)
- Ind-geometric stacks, with Sabin Cautis. (arXiv)
- Kasteleyn operators from mirror symmetry, with David Treumann and Eric Zaslow. (arXiv) (Selecta Math.)
- Cluster theory of the coherent Satake category, with Sabin Cautis. (arXiv) (J. Amer. Math. Soc.)
- Affine cluster monomials are generalized minors, with Dylan Rupel and Salvatore Stella. (arXiv) (Compos. Math.)
- On generalized minors and quiver representations, with Dylan Rupel and Salvatore Stella. (arXiv) (Int. Math. Res. Not.)
- On the combinatorics of exact Lagrangian surfaces, with Vivek Shende and David Treumann. (arXiv)
- Cluster varieties from Legendrian knots, with Vivek Shende, David Treumann, and Eric Zaslow. (arXiv) (Duke Math. J.)
- The greedy basis equals the theta basis, with Man-Wai Cheung, Mark Gross, Greg Muller, Gregg Musiker, Dylan Rupel, Salvatore Stella. (arXiv) (J. Combin. Theory Ser. A)
- Toda systems, cluster characters, and spectral networks. (arXiv) (Comm. Math. Phys.)
- Q-systems, factorization dynamics, and the twist automorphism. (arXiv) (Int. Math. Res. Not.)
- Cluster ensembles and Kac-Moody groups. (arXiv) (Adv. Math.)
- Double Bruhat cells in Kac-Moody groups and integrable systems. (arXiv) (Lett. Math. Phys.)

Expository writing:

- Notes on derived categories as higher categories. (link)

### Teaching

In Spring 2023 I am teaching both sections of MATH 430 "Theory of Numbers". Course information and communications are available through Blackboard.

### Other Stuff

I was previously a postdoctoral researcher at the Center for Quantum Mathematics and Physics at UC Davis. Before that I was supported by NSF Postdoctoral Research Fellowship DMS-1502845 at the University of Texas at Austin.

I organize the annual Los Angeles Workshop on Representations and Geometry (LAWRGe, /lärj/). LAWRGe 2024 will be held June 10-14, will be on "Relative Langlands Duality", and will be mentored by David Ben-Zvi, Yiannis Sakellaridis, and Akshay Venkatesh. You can find information about previous workshops here: 2022, 2023.

I like playing music, and as an undergraduate I studied classical guitar at the UT Austin School of Music.

I previously worked on machine learning and perception algorithms for self-driving cars as a research consultant for Helm.ai.

You can find more detailed biographical information in my CV.

### Contact Me

University of Southern California

Department of Mathematics

KAP 424B

3620 S. Vermont Ave

Los Angeles, CA 90089

hwilliams@usc.edu