Moduli spaces in enumerative and combinatorial geometry

研究者将研究代数几何中的几个计数和组合问题。 特别是，她将研究Gromov-Witten Theory of Deligne-Mumford stacks，并将其应用于经典的枚举几何，曲线模空间的高维类似物，以及Hessenbergvarieties和仿射Grassmannians的（等变）上同调和（等变）舒伯特演算。这些是不同的项目，但相似之处在于每个涉及的空间都有丰富的组合结构。 在代数几何中，理解满足一定几何条件的物体是一个基本问题，通常是由代数方案或堆栈自然参数化的。 这样的模和参数空间在 数学和科学研究的前沿。 枚举和组合问题的出现是特别令人兴奋的一部分碰撞领域的代数几何，代数组合，辛几何，和代表性理论，以及拓扑和理论物理。

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.