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:
- Contact isotopies in the coherent-constructible correspondence, with Jishnu Bose. (arXiv)
- Tropical Lagrangian coamoebae and free resolutions, with Christopher Kuo. (arXiv)
- 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.)
Notes:
Teaching
In Spring 2025 I am teaching two sections of MATH 407 "Probability Theory". 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 am an organizer for the 7th Canada-Mexico-US Meeting on Representation Theory, Noncommutative Algebra, and Categorification, held at USC from July 28-31, 2025.
I organized the Los Angeles Workshop on Representations and Geometry (LAWRGe, /lärj/) for three years. You can find information about these workshops here: 2022, 2023, 2024.
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