Mathematical Sciences: Anti Self-dual Connections on Algebraic Surfaces

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This award supports the research in Algebraic Geometry of Professor David Gieseker of the University of California at Los Angeles. Dr. Gieseker's investigations will be in the isospectral theory of difference operators, considering the Picard group of the Bloch manifold of a two-dimensional periodic difference operator. He will also explore generalizations of the Bloch manifold construction to other contexts such as the triangular- ization of a surface of arbitrary genus. This is research in the field of Algebraic Geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from Algebra, but from Analysis and Topology, and conversely is finding application in those fields as well as in theoretical Computer Science, Robotics, and Physics.