Enumerative geometry of moduli spaces and applications

该建议涉及代数几何中的问题，即研究多项式方程的解和此类解集的几何性质。PI研究的一个共同主题是研究几何对象的参数空间和有关它们的计数问题(例如，计算多少次二次多项式满足某些性质)。虽然这些问题一开始是无害的，但它们很快就变得复杂起来，现代方法涉及到几何思想与数学和物理其他领域的技术和猜想的结合。例如，在提出的研究课题之一中，PI计划研究某些参数空间与纽结理论中的问题之间的关系；在以前的工作中，PI受到数学物理思想的启发，证明了这种关系的一个特例。在其他研究课题中，PI将研究类似的现象；在每一种情况下，人们都希望在两个方向上都能得到富有成效的反馈，并希望研究这些参数空间的新技术将作为结果而发展。除了这项建议的研究方面外，国际数学教育协会计划在不同的层面上为数学教育提供支持。计划中的支持包括对初中和高中女性的支持，与美国数学联合开展的活动，研究生水平的课程和暑期讲座。这项提议的重点是研究代数几何中各种对象(轮、曲线、曲面)的模空间的计数几何的主题，以及来自邻近领域的问题和应用。第一个主题是Donaldson-Thomas理论，其中提出的项目涉及基于消失圈的扩展技术，目标是证明该主题中长期存在的几何猜想。在曲线奇点和结点不变量的研究中也有一些应用。第二个主题是箭图簇的量子上同调；在这里，PI与A.Okounkov共同开展了一个长期项目，将几何问题与量子群的构造联系起来。第三个主题是特征p中的代数曲面，PI计划使用Noether-Lefschetz度的几何来研究族中圈的行为。在这种情况下，目标的动机是理解算术几何中泰特猜想的结果。

The proposal concerns questions in algebraic geometry, which is the study of solutions to polynomial equations and the geometric properties of the set of such solutions. A common theme of the PI's research is the study of parameter spaces of geometric objects and enumerative questions about them (e.g. counting how many degree 2 polynomials satisfy certain properties). While these questions start innocuously, they quickly become complicated and the modern approach involves a combination of geometric ideas with techniques and conjectures from other areas of mathematics and physics. For example, in one of the topics for proposed research, the PI plans to investigate the relation between certain parameter spaces and questions in knot theory; in previous work, the PI proved a special case of such a relation, motivated by ideas from mathematical physics. In the other research topics, the PI will study similar phenomena; in each case, one expects fruitful feedback in both directions and hopes that new techniques for studying these parameter spaces will develop as a consequence. In addition to the research aspects of this proposal, the PI plans to apply support towards mathematics education at different levels. Planned support includes outreach for middle and high-school women, activities joint with Math for America, and graduate-level courses and summer-school lectures.The focus of this proposal is to study topics in the enumerative geometry of moduli spaces of various objects in algebraic geometry (sheaves, curves, surfaces), as well as questions and applications coming from neighboring fields. The first topic is Donaldson-Thomas theory, where the proposed projects involve extending techniques based on vanishing cycles, with the goal of proving longstanding geometric conjectures in the subject. There are also proposed applications to the study of curve singularities and knot invariants. The second topic is quantum cohomology of quiver varieties; here the PI, jointly with A. Okounkov, has a long-term project relating geometric questions to constructions from quantum groups. The third topic is algebraic surfaces in characteristic p, where the PI plans to study the behavior of cycles in families, using the geometry of Noether-Lefschetz degrees. In this case, the objectives are motivated by understanding consequences of the Tate conjecture in arithmetic geometry.