Moduli spaces in enumerative and combinatorial geometry

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The investigator will study several enumerative and combinatorial problemsin algebraic geometry. In particular, she will study Gromov-Witten theoryof Deligne-Mumford stacks with applications to classical enumerativegeometry, higher-dimensional analogues of the moduli space of curves, and(equivariant) cohomology and (equivariant) Schubert calculus of Hessenbergvarieties and affine Grassmannians. These are distinct projects, but aresimilar in that each of the spaces involved has a rich combinatorialstructure. It is a fundamental problem in algebraic geometry to understand objectssatisfying 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 ofa collision of the fields of algebraic geometry, algebraic combinatorics, symplectic geometry, and representation theory, as well as topology and theoretical physics.