Enumerative geometry of moduli spaces and applications

该建议涉及的问题，代数几何，这是研究解决方案的多项式方程和几何性质的一套这样的解决方案。PI研究的一个共同主题是研究几何对象的参数空间和关于它们的枚举问题（例如计算有多少个2阶多项式满足某些性质）。 虽然这些问题开始时无害，但它们很快变得复杂，现代方法涉及几何思想与其他数学和物理领域的技术和知识的结合。例如，在其中一个提议的研究主题中，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证明了这种关系的一个特例，其动机是数学物理学的思想。 在其他研究课题中，PI将研究类似的现象;在每种情况下，人们都希望在两个方向上都有富有成效的反馈，并希望研究这些参数空间的新技术将因此而发展。 除了这项建议的研究方面，PI计划对不同层次的数学教育提供支持。计划中的支持包括对初中和高中妇女的宣传，与Math for America联合开展的活动，以及研究生课程和暑期学校讲座。该计划的重点是研究代数几何中各种对象（层、曲线、曲面）的模空间的枚举几何，以及来自邻近领域的问题和应用。 第一个主题是唐纳森-托马斯理论，其中提出的项目涉及基于消失循环的扩展技术，目的是证明该主题中长期存在的几何结构。 也有建议的应用程序的研究曲线奇异性和结不变量。 第二个主题是量子上同调的π品种;这里的PI，与A。Okounkov，有一个长期的项目，涉及几何问题的建设，从量子群。 第三个主题是特征p的代数曲面，PI计划使用Noether-Lefschetz度的几何来研究族中循环的行为。在这种情况下，这些目标的动机是理解后果的泰特猜想算术几何。

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.