Since the era of Newton, mathematics has been a key tool in helping us comprehend the nature of the universe. Famous examples are calculus via Newtonian mechanics and differential geometry via Einstein's theory of general relativity. During the last twenty years, there has been a great deal of activity devoted to building a so-called string-theoretic model of the universe, which incorporates some of the most sophisticated mathematics. The subject of Gromov-Witten theory was born twenty years ago during a period of intensive interaction between mathematics and physics. Since then Gromov-Witten theory has established itself as a central area in both geometry and physics. At the same time, it has expanded greatly in its scope to many diverse areas of mathematics, ranging from the classical topic of Hurwitz theory to the modern area of Donaldson-Thomas invariants (the sheaf-theoretic counterpart of Gromov-Witten theory). Despite its success, many central problems remain unsolved. Two notable examples are the computation of higer genus Gromov-Witten invariants of compact Calabi-Yau manifolds and the precise relation between Gromov-Witten and Donaldson-Thomas invariants. The resolution of these problems is of great importance for geometry and physics. In this proposal, a team of the best experts in the world is assembled to attack these central problems. In addition, the PIs propose to develop technology to study a variety of questions relating Gromov-Witten theory to enumerative algebraic geometry, symplectic geometry and mathematical physics. The PIs hope to make important and substantial contributions to these areas of mathematics, and their interrelations.
This project is interdisciplinary in nature, in that both physical and mathematical ideas play central roles. In this sense it adds to the current trend of interaction between mathematics and physics. This project emphasizes teamwork and collaboration. Through research seminars, organizing and participating in national and international conferences, this proposal will also enhance the training of undergraduate and graduate students, as well as postdoctoral fellows. There will be a number of research publications that will help in introducing students to this exciting area of mathematics.
This award is cofunded by the Algebra and Number theory and the Topology programs of DMS.