Mathematics

Research

pdf Some words of inspiration

Publications and Preprints

pdf Cell Decompositions of Hecke Traces and Link Polynomials
Extended abstract. Accepted to Proceedings of the 37th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC) (Sapporo), 2025.
+ N. Williams

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

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

arxiv The Hilb-vs-Quot Conjecture
To appear in J. reine angew. Math.
+ O. Kivinen

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

arxiv Unipotent Elements and Twisting in Link Homology

arxiv From the Hecke Category to the Unipotent Locus

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

web 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 Affine Springer Fibers and Level-Rank Duality. UChicago, 1/7/25.
• pdf Fock Spaces, Braid Varieties, and Block Equivalences. Northeastern, 10/31/24.
• pdf Character Formulas from Lusztig Varieties and Affine Springer Fibers. WUSTL, 9/30/24.
• pdf Triply-Graded Link Homology and the Hilb-vs-Quot Conjecture. IASM (Zhejiang), 1/22/23.
• 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

web 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 Uniformization of Principal Bundles. UChicago, 2/22/23.
• 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