Moduli Spaces Relative Singular Divisors and Lagrangians

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AbstractAward: DMS-0905738Principal Investigator: Eleny IonelThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The proposal is aimed at increasing understanding of thestructure of the moduli spaces of stable holomorphic maps bycombining ideas from several different fields of research. Thecore of the project seeks to use geometric and analytical methodsto investigate what happens to these moduli during certainnatural degenerations. In particular it leads to defining aversion of Gromov-Witten invariants relative certain type ofsingular subspaces, including relative both a normal crossingsymplectic divisor and a Lagrangian intersecting in a prescribedway. It also investigates how these invariants behave underappropriate smoothings of either the ambient space or of thedivisor and the Lagragian.The proposed work lies at the intersection of string theory andsymplectic topology. String theory developed as a potentialcandidate for unifying general relativity and particlephysics. The details of this theory have turned out to beextraordinarily rich, and have inspired many remarkable resultsin mathematics. But results in mathematics have also guided andinspired many new discoveries in string theory and mirrorsymmetry. It is hoped that this project will contribute to thegrowing interaction between various fields of mathematics andother sciences. In particular, one of the themes running throughthis proposal is that symplectic topology can contribute insightsinto string theory.