Moduli spaces in enumerative and combinatorial geometry

研究者将研究代数几何中的几个枚举和组合问题。特别是，她将研究Deligne-Mumford堆栈的Gromov-Witten理论及其在经典枚举测量学中的应用，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）Schubert演算。这些是不同的项目，但相似之处在于每个空间都有丰富的组合结构。理解满足某些几何准则的对象是代数几何中的一个基本问题，这些准则通常是由代数方案或代数堆栈自然参数化的。这样的模和参数空间是数学和科学研究的前沿。作为代数几何、代数组合学、辛几何、表示理论以及拓扑学和理论物理领域碰撞的一部分，出现的枚举和组合问题特别令人兴奋。

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.