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.
