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This proposal would support research by three related groups of
researchers into the frontiers of Representation Theory, Number
Theory, and Mathematical Physics. The Representation Theory group
will compute the Kazhdan-Lusztig-Vogan polynomials for all simple Lie
groups up to rank 9 and make the results of these computations readily
available to researchers worldwide. This group will also continue to
explore the combinatorial infrastructure of W-graphs and relate it to
representation theoretical invariants. The Number Theory group will
carry out major computations of modular forms and L-functions, and
greatly enhance our understanding of the Birch and Swinnerton-Dyer
conjecture and the Riemann Hypothesis, two of the central problems in
number theory. The Mathematical Physics group will complete the first
major step in the classification of off-shell representations of
Supersymmetry, a problem that has been open for three decades.
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