RUI:Curve Counting Theories and Their Correspondences

San Francisco State University, San Francisco CA

Investigators

Abstract

Enumerative geometry is a classical project, encompassing questions that date back to antiquity (how many conics, for example, pass through five given points in the plane?) as well as subjects of contemporary research. Despite their long history, many of these questions remained inaccessible until a series of breakthroughs in the late twentieth century that culminated in the development of Gromov-Witten theory. The key new ideas were that problems of enumerative geometry are best interpreted in terms of intersection theory on a moduli space, and that insights from the physics of string theory reveal elegant patterns in families of enumerations that were previously unobserved by the mathematics community. The PI's research program involves developing the intersection theory of a particularly fundamental moduli space - the Deligne-Mumford moduli space of curves - as well as producing rigorous mathematical frameworks for some of the correspondences between theories that physicists have predicted. A major part of this research program is devoted to studying the Chow ring of the moduli space of curves. Although a full picture of the Chow ring of this space currently seems out of reach, there is a subring known as the "tautological ring" that contains nearly every geometrically-interesting class yet that admits an explicit, finite set of additive generators. The PI has been an active contributor to the study of the relations among these generators, in joint work with Felix Janda (University of Michigan), Sam Grushevsky (Stony Brook University), and Dmitry Zakharov (Central Michigan University). In forthcoming work, she plans to develop tautological intersection theory both computationally (producing algorithms for calculating certain desirable expressions for tautological classes in terms of the generators) and theoretically (seeking new tautological expressions for classes such as the hyperelliptic locus). Much of this work will be carried out in collaboration with student researchers. A second central component of the PI's research is the mathematical proof and extension of equivalences proposed by physics. For example, in joint work with Felix Janda and Yongbin Ruan (University of Michigan), she has proven a wall-crossing formula relating Gromov-Witten theory to the theory of quasimaps, a relationship that is closely related to the famous physical phenomenon known as mirror symmetry. In future work, she will extend this wall-crossing formula to new cases and use it to attack another physical conjecture: the Landau-Ginzburg/Calabi-Yau correspondence. She will also work, in collaboration with Alexandr Buryak (University of Leeds) and Ran Tessler (ETH Zurich), toward the theoretical development of a version of r-spin theory for curves with boundary, with an ultimate goal of generalizing Witten's r-spin conjecture to that setting. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.