Some problems in algebraic geometry with connections to string theory

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A series of problems are proposed in algebraic geometry, inspired bystring theory. Some problems in string theory relating to algebraicgeometry are also proposed. These include the study ofalgebro-geometric invariants related to topological string amplitudes:Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants, allconjecturally related to each other. Particular problems include arigorous definition of Gopakumar-Vafa invariants and theirgeneralization from Calabi-Yau threefolds to more general threefolds,their computation in examples, and verification of the GW-GVcorrespondence, beginning with toric varieties and contractiblecurves. These invariants will be extended to threefolds which are notCalabi-Yau. Computational methods for Gromov-Witten andDonaldson-Thomas invariants are proposed. Several connections tophysics will be pursued, including black hole entropy techniques.The project is a continuation of the exciting process ofcross-fertilization between string theory in physics and geometry inmathematics. In string theory, the fundamental particles of natureare small vibrating strings, and physical properties depend on thegeometry of the space that strings propagate in, much as the shape ofa drum affects the sound that it makes. Physically relevantproperties can be determined by calculations in pure geometry, in thisinstance calculations of intrinsic interest in mathematics. Whenphysical methods of duality provide more than one mathematical model,a consequence is a deep unexpected insight into geometry. In this project,the PI will develop tools to investigate the mathematical validity ofsome of these predictions and will expand the geometric investigationbeyond the physically relevant context. The PI will apply alsogeometric techniques to advance understanding of string theory,especially as it describes the statistical mechanics of a black hole.The primary mathematical content is the counting of curves in curvedspaces called Calabi-Yau threefolds which form the small extradimensions in string theory.