Research

My research is between algebraic geometry and algebraic combinatorics. Here are some thumbnail sketches of fields where these fields collide.

  • Geometric representations of the symmetric group. Particularly nice algebraic varieties carry an action of the permutation group on their cohomology. Springer varieties are a famous example: the top dimensional cohomology of Springer varieties gives each irreducible representation of the symmetric group.
  • Modern Schubert calculus. Classical Schubert calculus discovered that three apparently unrelated quantities were in fact the same: intersections of linear subspaces of a vector space, structure constants in the cohomology ring of a variety called the Grassmannian, and tensor product multiplicities of irreducible representations of the group of invertible n x n matrices. Modern Schubert calculus asks similar questions about different Lie types or cohomology theories, using more combinatorics and geometry.
  • Equivariant cohomology using combinatorial methods. Goresky, Kottwitz, and MacPherson introduced a combinatorial algorithm to calculate equivariant cohomology of a suitable complex projective variety, often called GKM theory. GKM theory turns a variety into a labelled combinatorial graph; an algebraic algorithm computes cohomology from the graph.

My projects often involve Hessenberg varieties, a family of subvarieties of the flag variety that include Springer varieties. This page describes Hessenberg varieties in detail and gives tables with their Betti numbers.

Selected current projects

  • Calculating the equivariant cohomology of weighted projective space using GKM theory. This was inspired by recent work of Bahri, Franz, and Ray.
  • Using Khovanov's model for the (n,n) Springer variety to give a geometrically and combinatorially explicit of Springer's representation. As an application, we decompose Springer representations into irreducibles in each degree, and give a new basis for Specht modules for two-row partitions. This is joint with Heather Russell.

Publications and preprints

Disclaimer: The arXived version of each paper is not necessarily the final version; it may have more typos than the published paper.

Submitted

  1. Schubert polynomials and classes of Hessenberg varieties, with Dave Anderson. Available at arXiv:0710.3182.
  2. A sharp diameter bound for unipotent groups of classical type over Z/pZ, with Jordan Ellenberg. Available at arXiv:math/0510506.
  3. Divided difference operators for partial flag varieties.

Accepted

  1. Permutation actions on equivariant cohomology, to appear in the Toric Topology volume of Contemporary Mathematics (Proceedings of the conference at Osaka City University, Japan). Available at arXiv:0706.0460.
  2. Permutation representations on Schubert varieties, to appear in the American Journal of Mathematics. Available at arXiv:math/0604578.

Published

  1. Hessenberg varieties are not pure dimensional, Journal of Pure and Applied Mathematics Quarterly 2 (2006), 779-794. Special issue dedicated to Robert MacPherson's 60th birthday. Available at arXiv:math/0601681.
  2. An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson, an expository paper presented at the 2004 Snowbird conference for young algebraic geometers and published in Snowbird lectures in algebraic geometry, 169-188, Contemp. Math. 388, Amer. Math. Soc., Providence, RI, 2005. Available at arXiv:math/0503369.
  3. Paving Hessenberg varieties by affines, Selecta Mathematica (N.S.), 13 (2007), 353-367. Available at arXiv:math/0409118.
  4. Linear conditions imposed on flag varieties, American Journal of Mathematics 128 (2006), 1587-1604. Available at arXiv:math/0406541.
  5. Exponents for B-stable ideals, with Eric Sommers, Transactions of the American Mathematical Society 358 (2006), 3493-3509. Available at arXiv:math/0406047.
  6. Distinguishing numbers for graphs and groups, Electronic Journal of Combinatorics 11 (1) 2004, #R63. Available at arXiv:math/0406542.
  7. Decomposing Hessenberg varieties over classical groups, Ph. D. thesis. Available at arXiv:math/0211226.