Moduli spaces in enumerative and combinatorial geometry

研究人员将研究代数几何中的几个计数和组合问题。特别是，她将学习Deligne-Mumford堆栈的Gromov-Witten理论及其在经典计数计量学、曲线模空间的高维类似以及Hessenberg变种和仿射Grassmannians的(等变)上同调和(等变)Schubert演算中的应用。这些都是不同的项目，但都是相似的，因为每个涉及的空间都有丰富的组合结构。理解满足某些几何标准的对象是代数几何中的一个基本问题，这些标准通常由代数方案或堆栈自然地参数化。这种模和参数空间处于数学和科学研究的前沿。作为代数几何、代数组合学、辛几何和表示论以及拓扑学和理论物理学领域碰撞的一部分，出现的计数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.