Mathematics

Research

Publications and Preprints

arxiv Free Immersions in Critical Dimension from Lie Groups

arxiv Haiman’s Conjecture and Springer’s Representations
Here are accompanying data and scripts.

arxiv Partial Resolutions and Noncrossing Combinatorics
+ N. Williams

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 errata.
+ O. Kivinen

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

Extended Abstracts

arxiv Level-Rank Dualities for Finite Reductive Groups
To appear in Math. Res. Lett. (2026).
+ T. Xue

web Cell Decompositions of Hecke Traces and Link Polynomials
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

Selected Slides

• pdf Haiman’s Conjecture and the Springer Correspondence. MIT, 4/29/26.
• 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