Gromov-Witten theory is a rapidly expanding field with basic connections to many central areas of current research in mathematics and physics. The project proposed here is a wide ranging study of Gromov- Witten theory based on the techniques and discoveries of the last few years. The main topics covered are: the definition and exact evaluations of integrals on the moduli space of curves with boundaries, the proof of the universal Virasoro constraints, the establishment of the Gromov-Witten/ Donaldson-Thomas/Pairs correspondences, and the study of tautological classes. These topics point in several different directions: topological string theory, integrable hierarchies, and classical algebraic geometry. Each topic is central to progress in the field, and each will be addressed with a new point of view.
Algebraic varieties, defined by the zeros of polynomial equations, are basic objects in both classical and modern mathematics. Algebraic geometry is the study of algebraic varieties. Ideas from symplectic geometry and string theoretic physics have opened new fields in algebraic geometry: the study of algebraic varieties via the Gromov-Witten theory of their curves and the Donaldson-Thomas theory of their sheaves. Since the topic has basic connections in several directions, progress will have a direct impact on the neighboring fields. Topological string theory is the most obvious connection and the two fields are in frequent contact. But also, for example, topics varying from the Fukaya category to random 3-dimensional partitions will be affected.
Algebraic geometry is the study of the zero sets of polynomial equations in several variables. The subject has a central role in mathematics with connections to number theory, representation theory, and topology. Moduli questions in algebraic geometry concern the behavior of the solution set as the coefficients of the polynomials vary. At the end of the 20th century, several fundamental links between the algebraic geometry of moduli problems and path integrals in quantum field theory were made. The subject today uses insights and techniques with origins in both mathematics and physics. The grant supported my research in several directions related to the geometry of moduli spaces.The principal discoveries are: (i) In a series of papers with Richard Thomas, we developed the theory of stable pairs. A stable pair is a sheaf supported on a curve with a section. The moduli of stable pairs leads to a very accessible approach to Donaldson-Thomas theory. (ii) In a series of papers with Aaron Pixton, we studied the descendent theory of stable pairs proving, among several properties, the rationality of the associated generating series. After sufficient understanding of this theory, we proved the Gromov-Witten/Donaldson-Thomas correspondence for many compact Calabi-Yau 3-folds (conjectured in 2003). The latter result provides new structure to the Gromov-Witten theory of Calabi-Yau quintic 3-fold. (iii) In a series of papers with Aaron Pixton (and later Dmitri Zvonkine), we study the tautological ring of the moduli space of curves. With Pixton, we proved the Faber-Zagier conjecture of 2000 proposing an explicit formula for relations in the tautological ring. After Pixton extended the Faber-Zagier formula over the Deligne-Mumford compactification, with Pixton and Zvonkine, we proved the full extension yields relations. These results have changed the perspective of the field. (iv) In the Gromov-Witten theory of K3 surfaces, I have proven (with various coauthors) the Yau-Zaslow conjecture for all classes and the Katz-Klemm-Vafa conjecture (with Richard Thomas) for all genera and classes. These both concern the geometry of curves on K3 surfaces and were conjecture in 1995 and 1999 respectively.
