Mathematics

Research

Publications and Preprints

Central Elements and Cell Decompositions from Partial Springer Resolutions
+ N. Williams

arxiv Level-Rank Dualities for Finite Reductive Groups
Extended abstract.
+ T. Xue

arxiv Level-Rank Dualities from \(\Phi\)-Cuspidal Pairs and Affine Springer Fibers
+ T. Xue

arxiv Unipotent Elements and Kálmán–Serre Duality

arxiv From the Hecke Category to the Unipotent Locus

doi arxiv The Hilb-vs-Quot Conjecture
J. reine angew. Math. (Crelle), 828 (2025), 83–126. Here is a list of minor errata.
+ O. Kivinen

web Cell Decompositions of Hecke Traces and Link Polynomials
Extended abstract. Proceedings of the 37th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), Sapporo, 2025. Sém. Lothar. Combin., 93B (2025), Paper No. 90.
+ N. Williams

doi arxiv Simple Braids Tend toward Positive Entropy
J. Knot Theory Ramifications, 33(11) (2024), 2450034, 12 pp.
+ L. Robitaille

doi arxiv Rational Noncrossing Coxeter–Catalan Combinatorics
Proc. Lond. Math. Soc., 129(4) (2024), e12643, 50 pp.
+ P. Galashin, T. Lam, N. Williams

doi Extremal Primes for Elliptic Curves
J. Num. Theory, 164 (2016), 282–298.
+ K. James, B. Tran, P. Wertheimer, D. Zantout

doi arxiv Zeros of Dirichlet \(L\)-Functions over Function Fields
Comm. Num. Theory Phys., 8(3) (2014), 511–539.
+ J. Andrade, S. J. Miller, K. Pratt

Selected Slides

• pdf Hilb vs Quot vs HOMFLYPT. UMaryland, 12/1/25.
• pdf Zeta Functions as Knot Invariants. Howard Colloquium, 9/19/25.
• pdf Affine Springer Fibers and Level-Rank Duality. UChicago, 1/7/25.
• pdf Character Formulas from Lusztig Varieties and Affine Springer Fibers. WUSTL, 9/30/24.
• pdf Catalan Combinatorics in Algebraic Geometry. MIT, 2/15/23.
• pdf Braids, Unipotent Representations, and Nonabelian Hodge Theory. Oxford, 11/30/21.
• pdf Homotopy Equivalences of Varieties Built from Braids. UMass Amherst, 10/25/21.
• pdf From \(\mathbf{H}\) to \(\mathcal{U}\). UMass Amherst, FRG Conference, 6/17/21.

Other Writing

Publications

doi The Hodge Theory of Soergel Bimodules
Chapter 18 in Introduction to Soergel Bimodules. Ed. B. Elias, S. Makisumi, U. Thiel, G. Williamson. RSME Springer Series, 5 (2020), 347–367.
+ L. Taylor

Selected Slides

• pdf Knots, Plethysms and the Riordan Group. Howard, 10/31/25.
• pdf Higgs Bundles and Global Springer Theory. Edinburgh, 3/15/22.
• pdf What Gauss Knew about Knots and Braids. MIT, IAP Lecture Series, 1/20/21. Here are the accompanying problems and references.

Teaching