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\LARGE\bf
SCREMS: The Frontiers of Representation
Theory, Number Theory, and Mathematical Physics
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\mypar{}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.
All three groups have worked very productively together during the
last two years, and if funded this proposal will likely lead to
many similar future collaborations.
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\noindent{\bf Intellectual Merit:}
\mypar{}Each of these three projects will advance the state of the art
in applications of computation to research mathematics, and
lead to new tools and data to support work in representation
theory, number theory, and mathematical physics.
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\noindent{\bf Broader Impact:}
\mypar{}All data and software that comes out of this project will be
made freely available over the Internet. This will result in tools
that will transform how researchers in mathematics share and
manipulate their data and collaborate. Much of the data that arises
out of these projects will also be of great value to many
mathematicians.
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