Quantum Cohomology, Quantum K-theory, Frobenius Manifolds and Integrable Systems

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AbstractAward: DMS-0305895Principal Investigator: Yuan-Pin LeeThe main focus of this project is to study Gromov--Witten theoryand its relations and applications to other areas of mathematics,including enumerative geometry, integrable systems, K-theory, andrepresentation theory. The emphasis is placed on some keyproblems where these areas intersect. These problems include thestudy of higher spin curves, quantum K-theory, enumerativegeometry, and Frobenius manifolds.Gromov--Witten theory was born in early 1990's through theinteraction of mathematics and physics. On the mathematical side,it starts with Gromov's invention of a new way to producesymplectic invariants. On the physical side, it was Witten'sstudy of string theory. Although of a relatively short history,Gromov--Witten theory is a major active field of research. Thisis due not only to its powerful solutions to old problems, butalso to its interdisciplinary nature, with continuing inputs fromseveral fields of mathematics and string theory. This projectproposes to investigate Gromov--Witten theory through variousperspectives and to search for its applications in other subjectareas.