Gromov-Whitten Theory and its Relations With K-Theory, Integrable Systems and Enumerative Geometry

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DMS-0072547Yuan-Pin LeeQuantum cohomology is a deformation of usual cup productof ordinary cohomology ring of a smooth projective variety X. Similarly quantum K-theory is a deformation of ordinary K-ring of X. In their study of Gromov--Witten K-theory, Professor Lee and his collaborators have found some interesting relations to other fields in mathematics and physics including algebraic geometry, integrable systems, representation theory and quantum geometry. These investigations also naturally lead to further investigations of Gromov--Witten theory itself. The main focus of this project will be on quantum cohomology, quantum K-theory and their relations to discrete KdV hierarchies, Toda lattices, and enumerative geometry. Gromov-Witten theory is a new subject that lies within theintersection of many traditional branches of mathematics and theoretical physics. This theory was originally discovered within string theory. Shortly thereafter people found many new applications to mathematics. Some of these applications solvedproblems that, based on traditional methods, were considered tobe very difficult. The tools used to study Gromov--Witten theory also involves many fields, including algebraic geometry, topology and analysis. Thus, this is a growing new field which exhibits many rich structures and deserves further investigation