About

Tom Braden is a full professor and the director of the Graduate Program in the Department of Mathematics and Statistics at UMass Amherst. His research focuses on the topology of singular algebraic varieties, particularly those with a combinatorial nature such as toric varieties and Schubert varieties. He employs invariants like intersection cohomology to analyze the combinatorial structure of polytopes and Coxeter groups derived from these varieties. A major theme in his work involves lifting or 'categorifying' relations and inequalities among combinatorial invariants by demonstrating that they originate from canonical relations among topological invariants. For example, an inequality may be shown to arise from the fact that a certain map of cohomology groups is an injection. He studies these relations by working equivariantly, leveraging the symmetries imposed by large groups acting on the varieties.

Research topics

• Pure mathematics
• Mathematics
• Geometry
• Discrete mathematics
• Combinatorics

Selected publications

• Intersection cohomology without spaces

  ArXiv.org · 2025-10-10

  preprintOpen access1st authorCorresponding

  We survey three settings in which dimensions of intersection cohomology groups of algebraic varieties provide deep combinatorial and representation-theoretic information, and computations of the groups themselves have been made using combinatorial sheaves on finite posets. These settings are (1) intersection cohomology of Schubert varieties, the associated Kazhdan-Lusztig polynomials and their realizations via moment graph sheaves and Soergel bimodules; (2) intersection cohomology of toric varieties, the associated g-polynomials of convex polytopes, and their realization via the theory of intersection cohomology of fans; and (3) intersection cohomology of arrangement Schubert varieties, the associated Kazhdan-Lusztig polynomials of matroids, and their realization via intersection cohomology of matroids. In all three settings these constructions are valid in more general situations where the variety does not exist, leading to "intersection cohomology without spaces." We give parallel presentations of these three stories, highlighting applications to KLS-polynomials.

  PublisherOA PDFDOI

• What is... the Dowling–Wilson Conjecture?

  Notices of the American Mathematical Society · 2024-09-12

  articleOpen access1st authorCorresponding
  PublisherOA PDFDOI

• Perverse sheaves on symmetric products of the plane

  arXiv (Cornell University) · 2022-08-30

  preprintOpen access1st authorCorresponding

  For any field $k$, we give an algebraic description of the category $\mathrm{Perv}_\mathscr{S}(S^n (\mathbb{C}^2),k)$ of perverse sheaves on the $n$-fold symmetric product of the plane $S^n(\mathbb{C}^2)$ constructible with respect to its natural stratification and with coefficients in $k$. In particular, we show that it is equivalent to the category of modules over a new algebra that is closely related to the Schur algebra. As part of our description we obtain an analogue of modular Springer theory for the Hilbert scheme $\mathrm{Hilb}^n(\mathbb{C}^2)$ of $n$ points in the plane with its Hilbert-Chow morphism.

  PublisherOA PDFDOI

• A semi-small decomposition of the Chow ring of a matroid

  Advances in Mathematics · 2022 · 39 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Combinatorics
  PublisherDOI

• A semi-small decomposition of the Chow ring of a matroid

  arXiv (Cornell University) · 2020 · 13 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Combinatorics

  We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations for the Chow ring, recovering the main result of [AHK18]. We also show that a similar semi-small orthogonal decomposition holds for the augmented Chow ring of M.

  PublisherOA PDFDOI

• Singular Hodge theory for combinatorial geometries

  arXiv (Cornell University) · 2020 · 30 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Discrete mathematics

  We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy conjecture and the nonnegativity of the coefficients of Kazhdan-Lusztig polynomials for all matroids.

  PublisherOA PDFDOI

• Kazhdan-Lusztig Polynomials of Matroids Under Deletion

  The Electronic Journal of Combinatorics · 2020-01-24 · 2 citations

  preprintOpen access1st authorCorresponding

  We present a formula which relates the Kazhdan–Lusztig polynomial of a matroid $M$, as defined by Elias, Proudfoot and Wakefield, to the Kazhdan–Lusztig polynomials of the matroid obtained by deleting an element, and various contractions and localizations of $M$. We give a number of applications of our formula to Kazhdan–Lusztig polynomials of graphic matroids, including a simple formula for the Kazhdan–Lusztig polynomial of a parallel connection graph.

  PublisherOA PDFDOI

• Quantizations of conical symplectic resolutions I: local and global structure

  Astérisque · 2018-11-06 · 45 citations

  article1st authorCorresponding
  PublisherDOI

• Ringel duality for perverse sheaves on hypertoric varieties

  Advances in Mathematics · 2018-12-20 · 2 citations

  preprintOpen access1st author
  PublisherOA PDFDOI

• Quantizations of conical symplectic resolutions II: category $\mathcal {O}$ and symplectic duality

  Astérisque · 2018-11-06 · 99 citations

  article1st authorCorresponding

  We define and study category $\mathcal O$ for a symplectic resolution, generalizing the classical BGG category $\mathcal O$, which is associated with the Springer resolution. This includes the development of intrinsic properties parallelling the BGG case, such as a highest weight structure and analogues of twisting and shuffling functors, along with an extensive discussion of individual examples. We observe that category $\mathcal O$ is often Koszul, and its Koszul dual is often equivalent to category $\mathcal O$ for a different symplectic resolution. This leads us to define the notion of a symplectic duality between symplectic resolutions, which is a collection of isomorphisms between representation theoretic and geometric structures, including a Koszul duality between the two categories.

  PublisherDOI

Frequent coauthors

• Nicholas Proudfoot

  19 shared

• Ben Webster

  13 shared

• Anthony Licata

  Australian National University

  13 shared

• Carl Mautner

  7 shared

• Linda Chen

  6 shared

• Frank Sottile

  6 shared

• Robert MacPherson

  Institute for Advanced Study

  6 shared

• Artem Vysogorets

  New York University

  5 shared