Mathematical Sciences: The Topology of Varieties

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Hain intends to use the techniques of Hodge theory to study the topology of the space of compact Riemann surfaces and the topology of the space of n x m matrices over the complex numbers, all of whose minors are non-zero. Possible applications include the construction of generalizations of Euler's dilogarithm which satisfy natural generalizations of Abel's 5-term functional equation for the dilogarithm. A graduate student will employ a computer program Hain has written to aid in investigating the dilogarithm. The program decomposes representations of symmetric groups, which arise as huge tensor products. A high-powered personal computer will be devoted to this task.