Harmonic analysis and global invariants

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

ABSTRACT: One of the themes visible in recent developments in mathematics is the cross-fertilization of ideas across different mathematical disciplines. As an example I mention interplay between topology and analysis in the computation of topological properties of a manifold. Another example involving number theory and topology and analysis can be found in the explanation of the leading analytic behavior of L-functions from number theory in terms of topological information from periods of topological cycles computed using harmonic analysis. Similar interdisciplinary research has been the main focus of recent research of the PI. Several projects have been concerned with the evaluation or determination of topological/geometrical properties of locally symmetric manifolds either with harmonic analysis techniques or through leading behaviour of geometric zeta functions. This theme continues in the present proposal. Joint projects have been begun to extend from Hermitian locally symmetric spaces to their "real forms" the construction of geometric zeta functions, and the identification of their leading behaviour in geometric terms. Part of this work involves developing refined information concerning various compatifications of symmetric spaces and then encapsulating it in a generating function of geometric origins. It is expected that torsion-like invariants will appear in this investigation but it is hoped that a better understanding of subtle characteristic classes like the R-class may result. A new direction taken by the PI is toward nilpotent orbits in semisimple Lie algebras. An approach to an algebraic proof of the existence of hyperKahler metrics on these orbits is underway. Also a realization of the quantization of the minimal orbit by means of Dirac operators has been started. The use of techniques from twistor theory play an important role in these investigations.