Principal Investigator: Eleny-Nicoleta Ionel
This proposal aims to understand the structure of the Gromov-Witten invariants of symplectic manifolds by combining together ideas coming from different fields of research. The first project is motivated by a conjecture made by two physicists, Gopakumar and Vafa. The conjecture implies several surprising restrictions on the Gromov invariants of a Calabi-Yau 3-fold. The goal is to prove the conjectured formula by adapting some analytical techiques developed by Taubes to relate the Seiberg-Witten and Gromov invariants in 4-dimensions. The second project's goal is to obtain new relations in the cohomology ring of the moduli space of complex structures on a marked Riemann surface. Using the techniques introduced in a previous paper, the PI found several families of interesting relations, one of which proves a ten year old conjecture of Faber. The last project seeks to extend the sum formula for Gromov-Witten invariants to deformations more general then those appearing from a symplectic sum. There already seems to be several interesting new phenomena appearing in the general case.
The proposed work lies at the intersection of string theory and symplectic topology. String theory developed as a potential candidate for a unifying theory of the universe, which extends Eistein relativity theory. It is based on the idea that elementary particles (like electrons, photons) should be thought not as points, but rather small vibrating loops. Working out the details of this theory turned out to be quite delicate, and has in turn inspired many remarkable results in mathematics. But also fundamental results in mathematics have inspired many new discoveries in physics. It is hoped that this project will contribute to the increased interaction between mathematics and high energy physics. In the same time, one of the projects will involve graduate students.
