Harold Williams

Harold Williams is an Assistant Professor in the Department of Mathematics at the University of Southern California, supported by an NSF CAREER Grant (DMS-2143922). His research bridges representation theory, geometry, and combinatorics with theoretical physics. He previously held a NSF Postdoctoral Fellowship (DMS-1502845) at the University of Texas at Austin and was a researcher at the Center for Quantum Mathematics and Physics (QMAP) at UC Davis.

Research Interests:

• Representation Theory
• Geometric and Categorical Structures in Quantum Field Theory
• Cluster Algebras and their Applications
• Symplectic Geometry and Mirror Symmetry
• Mathematical Physics (Integrable Systems, Gauge Theory)

Key Research Trends: His work emphasizes categorical and geometric approaches to algebraic structures, with notable contributions to cluster varieties, Coulomb branches, and Langlands duality. Recent publications explore connections between Kasteleyn operators, spectral networks, and mirror symmetry.

Awards and Grants:

• NSF CAREER Grant (2021–present)
• NSF Postdoctoral Fellowship (2015–2018)

Teaching and Outreach: In Spring 2025, he teaches MATH 407 'Probability Theory' at USC. He organizes the 7th Canada-Mexico-US Meeting on Representation Theory (July 2025) and ran the Los Angeles Workshop on Representations and Geometry (LAWRGe) from 2022–2024.

Additional Background: Studied classical guitar at UT Austin School of Music. Previously consulted for Helm.ai on machine learning for autonomous vehicles.