Research in Several Complex Variables and Applications

Purdue University, West Lafayette IN

Investigators

Abstract

Proposal Number DMS-0203072 PI: Laszlo Lempert ABSTRACT This project has two components. One concerns complex analysis on infinite dimensional manifolds, and the proposed problems are mostly cohomological in nature. They center around the computation of Dolbeault and sheaf cohomology groups. Two types of manifolds will be investigated: loop spaces of finite dimensional compact complex manifolds and Stein manifolds. One (longer term) goal will be to prove a Hirzebruch--Riemann--Roch type index theorem on loop spaces, another to discover an appropriate class of sheaves for which cohomology can be proved to vanish on infinite dimensional Stein manifolds (generalization of the notion of coherent sheaves). I will also work on solving the inhomogeneous Cauchy--Riemann equations in pseudoconvex domains in Hilbert spaces. The finite dimensional component is about extending the concept of holomorphic motion from the complex plane to motions of higher dimensional complex manifolds and their subsets. This will hopefully provide a useful tool for higher dimensional complex dynamics. Mathematics in general elucidates the formal structures that underlie the exploration of our world. An example of such a structure is a function, which formalizes the notion of "dependence" among variable quantities. For instance temperature at some location on Earth depends on the latitude and the longitude of the location, but also on the altitude, and the time of the measurement: we have a function of four variables. In another context, the temperature of say a mole of a gas depends on the velocity of each molecule of the gas, thus on trillions and trillions of variables. Functions of such a huge number of variables are best understood by idealizing to infinitely many variables. These are the main objects of the proposed research: functions of infinitely many variables. The variables are complex rather than real numbers. Introducing complex variables typically streamlines problems, and having solved the complexified problem one can often go back and obtain, by restriction, a solution to the original real problem. The problems to be considered are partly motivated by various mathematical disciplines, and also by theoretical physics.