RUI: Moduli Spaces and Combinatorial Algebraic Geometry

研究人员将研究代数几何中的各种组合和计数问题。与这些问题基本相关的是模和参数空间及其上同调理论的研究，对结构组合丰富的对象的研究，以及退化技术及其应用。特别是，研究者建议研究等变量子上同调理论、仿射Grassmannians、Hessenberg簇、FLAG簇、曲线的模空间、Quot方案、GKM空间和堆栈及其环退化。所提出的研究问题涉及代数几何的现代方法与组合学、辛几何和物理学的新发展之间的富有成效的互动。拟议的项目对加强对代数几何中基本对象及其相互之间的联系的理解具有重要的潜在影响。通过研究、教育、建议和组织，调查员将促进高校师生之间的互动和合作。

The investigator will study various combinatorial and enumerative problems in algebraic geometry. Fundamentally linked to these problems are the study of moduli and parameter spaces and their cohomology theories, the study of objects combinatorially rich in structure, and degeneration techniques and their applications. In particular, the investigator proposes to investigate equivariant quantum cohomology theories, affine Grassmannians, Hessenberg varieties, flag varieties, moduli spaces of curves, quot schemes, GKM spaces, and stacks and their toric degenerations.The proposed research problems are concerned with fruitful interactions between modern methods in algebraic geometry and new developments in combinatorics, symplectic geometry, and physics. The proposed projects have significant potential impact on enhancing the understanding of fundamental objects in algebraic geometry and their connections to one another. Through research, education, advising, and organizing, the investigator will promote interactions and collaborations between students and faculty of colleges and universities.