Gromov Witten Invariants of Singular Spaces

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AbstractAward: DMS-0605003Principal Investigator: Eleny-Nicoleta IonelThe proposal is aimed at increasing understanding of thestructure of the Gromov-Witten invariants of symplectic manifoldsby combining together geometric, topological and analyticalmethods. The first project aims to define the Gromov-Witteninvariants of a symplectic manifold relative a singular subspace(with mild singularities, e.g. normal crossings) and to usethose invariants to prove a generalized symplectic sumformula. The project uses both geometric and analytical methodsto investigate what happens to the moduli spaces of holomorphicmaps during certain kinds of natural degenerations, revealingsurprising new features but also new challenges andcomplications. There are several interesting applications of thiswork, one of them presented as a separate project, which involvesthe idea of using a Donaldson divisor to give a simple geometricdefinition of the virtual fundamental cycle. The third projectis motivated by a conjecture made by two string theorists,R. Gopakumar and C. Vafa. The PI has been working with ThomasParker on a structure theorem for Gromov invariants in 6dimensions; this would have many of the same consequences as theGopakumar-Vafa Conjecture, and it ties in nicely with C. Taubesdeep work on the relation between Seiberg-Witten and Gromovinvariants in 4 dimensions.The proposed work lies at the intersection of string theory andsymplectic topology. String theory developed as a potentialcandidate for unifying general relativity and particle physics.The details of this theory have turned out to be extraordinarilyrich, and have inspired many remarkable results inmathematics. But results in mathematics have also guided andinspired many new discoveries in string theory. It is hoped thatthis project will have the broader impact of adding momentum tothe growing interaction between mathematicians and theoreticalphysicists. In particular, one of the themes running through allthe projects in this proposal is that symplectic topology cancontribute insights into string theory. The research and otheractivities of the PI also have impact on the education of nextgeneration of mathematicians. One of the proposed projects willinvolve graduate students, engaging them in cutting-edge researchearly in their graduate career. The PI has also been active inencouraging and guiding women graduate students and will continueto strongly support young women entering mathematics.