Julianna Tymoczko
Associate Professor of Mathematics
Burton Hall 314
(413) 585–3775
Email: jtymoczko AT smith dot edu
Research
My research is between algebraic geometry and algebraic combinatorics. Some places where these fields collide are:
- 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.