The investigator will study several enumerative and combinatorial problems in algebraic geometry. In particular, she will study Gromov-Witten theory of Deligne-Mumford stacks with applications to classical enumerative geometry, higher-dimensional analogues of the moduli space of curves, and (equivariant) cohomology and (equivariant) Schubert calculus of Hessenberg varieties and affine Grassmannians. These are distinct projects, but are similar in that each of the spaces involved has a rich combinatorial structure.
It is a fundamental problem in algebraic geometry to understand objects satisfying certain geometric criteria, often naturally parametrized by an algebraic scheme or stack. Such moduli and parameter spaces are at the forefront of mathematical and scientific research. Enumerative and combinatorial problems which arise are of particular excitement as part of a collision of the fields of algebraic geometry, algebraic combinatorics, symplectic geometry, and representation theory, as well as topology and theoretical physics.
