Principal Investigator: Eleny-Nicoleta Ionel
The proposal is aimed at increasing understanding of the structure of the Gromov-Witten invariants of symplectic manifolds by combining together geometric, topological and analytical methods. The first project aims to define the Gromov-Witten invariants of a symplectic manifold relative a singular subspace (with mild singularities, e.g. normal crossings) and to use those invariants to prove a generalized symplectic sum formula. The project uses both geometric and analytical methods to investigate what happens to the moduli spaces of holomorphic maps during certain kinds of natural degenerations, revealing surprising new features but also new challenges and complications. There are several interesting applications of this work, one of them presented as a separate project, which involves the idea of using a Donaldson divisor to give a simple geometric definition of the virtual fundamental cycle. The third project is motivated by a conjecture made by two string theorists, R. Gopakumar and C. Vafa. The PI has been working with Thomas Parker on a structure theorem for Gromov invariants in 6 dimensions; this would have many of the same consequences as the Gopakumar-Vafa Conjecture, and it ties in nicely with C. Taubes deep work on the relation between Seiberg-Witten and Gromov invariants in 4 dimensions.
The proposed work lies at the intersection of string theory and symplectic topology. String theory developed as a potential candidate for unifying general relativity and particle physics. The details of this theory have turned out to be extraordinarily rich, and have inspired many remarkable results in mathematics. But results in mathematics have also guided and inspired many new discoveries in string theory. It is hoped that this project will have the broader impact of adding momentum to the growing interaction between mathematicians and theoretical physicists. In particular, one of the themes running through all the projects in this proposal is that symplectic topology can contribute insights into string theory. The research and other activities of the PI also have impact on the education of next generation of mathematicians. One of the proposed projects will involve graduate students, engaging them in cutting-edge research early in their graduate career. The PI has also been active in encouraging and guiding women graduate students and will continue to strongly support young women entering mathematics.
