Enumerative geometry of moduli spaces and applications

该提案涉及代数几何中的问题，即多项式方程的解和这些解的集合的几何性质的研究。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，计算有多少次2次多项式满足某些性质）。虽然这些问题开始时很简单，但它们很快就变得复杂起来，现代的方法包括将几何思想与其他数学和物理领域的技术和猜想结合起来。例如，在提议的研究课题之一中，PI计划研究某些参数空间与结理论问题之间的关系；在之前的工作中，PI证明了这种关系的一个特殊情况，其动机来自数学物理的思想。在其他研究课题中，PI将研究类似的现象；在每一种情况下，人们都期望在两个方向上都有卓有成效的反馈，并希望研究这些参数空间的新技术将因此得到发展。除了这项建议的研究方面，PI计划在不同层次的数学教育中应用支持。计划提供的支持包括向初高中女性伸出援手，与“美国数学”联合开展活动，以及研究生课程和暑期学校讲座。本提案的重点是研究代数几何中各种对象（束、曲线、曲面）的模空间的枚举几何主题，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周期的扩展技术，其目标是证明该主题中长期存在的几何猜想。本文还提出了在曲线奇异性和结不变量研究中的应用。第二个主题是颤振变种的量子上同调；在这里，PI与a . Okounkov合作，有一个关于量子群构造的几何问题的长期项目。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的环的行为。在这种情况下，目标的动机是理解算术几何中的Tate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.