Professor Bloch will study relations between zeta functions and representations of infinite dimensional Lie algebras and function groups. He will focus on relations with values of zeta and L-series, quasi-modularity and related modular properties extending known modular properties for characters in one variable and relations with mirror symmetries. The investigator will also attempt to discover a Riemann Roch theorem for vector bundles with flat connections, using algebraic Chern Simons character classes.
This is a project in both representation theory and geometry. It is a common occurrence in mathematics that a investigation will produce an infinite sequence of numbers representing basic information about the objects under study. Such a sequence can appear to form quite randomly, even though it is actually completely determined by the way it was identified. Certain mathematical functions, zeta and L-series, have long been known to contain important information about the distribution of numbers in such sequences. Important examples of zeta and L-series are often connected to the concrete representations of the abstract mathematical constructions, algebras and groups. Unfortunately, mathematicians do not fully understand either the connection between number sequences and these series or the connection between the series and the representations. If both of these connections were regularized, mathematics would have a powerful new method of discovery. This project will investigate the second set of these connections.
