Automorphic Forms and Lie Algebras

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My research centers on automorphic forms and L-functions, and their applications in Number Theory and Lie Algebras. One of the most important mathematical ideas of the second half of the 19th century is that an analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular polynomial equation, but discovers that the solutions are very hard to find. On the other hand, one might want to count the number of solutions to a sequence of equations modulo prime numbers and have the answers as a sequence, namely, discrete information. One can form the so-called Artin L-function out of the discrete information. This L-function is an analytic function where one can apply analysis, and this L-function encodes the discrete information. Another example is that zeros of the Riemann zeta function provide the answer to the problem of counting prime numbers.Automorphic forms are certain analytic functions which have many symmetries. Finite dimensional simple Lie algebras are of fundamental importance in geometry and physics as a measure of symmetry. Since then, several generalized Lie algebras were introduced such as Kac-Moody algebras and Borcherds algebras. Amazingly, automorphic forms occur in these algebras and thus Number Theory and Lie Algebras became intimately related. My research will develop this relationship further.