Complex analysis and Geometry in Infinite Dimensions

Purdue University, West Lafayette IN

Investigators

Abstract

The project's focus is on complex analysis and geometry in infinite-dimensional manifolds. The latter topic especially represents largely uncharted territory. The principal investigator will explore it by considering fundamental results of the finite-dimensional theory and asking how they generalize to the infinite-dimensional setting. In some cases one already has a sense of what the generalization should be, and the challenge is actually to prove it; in other cases even the terms in which to formulate a generalization must be discovered. Two notions are central to the project. One is the concept of a complex loop space. Such a space is obtained by starting with a (finite-dimensional) complex manifold. The collection of all loops in this manifold is an infinite-dimensional complex manifold known as a complex loop space. The other central idea is that of a cohesive sheaf, an infinite-dimensional generalization of the all-important coherent sheaves of the finite-dimensional theory that the principal investigator and a collaborator have recently introduced. A large part of the project is the study of these two notions, especially their confluence: namely, cohesive sheaves over loop spaces, and their cohomology groups. The project will also seek applications in other areas of mathematics for the results that the project expects to obtain. The location of a point or point-mass in three-dimensional space can be described by its coordinates, that is, by three numbers. Accordingly, since Descartes we have known that curves and surfaces in space can be described by functions of three variables. Now to specify the position of a more complicated object, say a Frisbee, the three coordinates of its center of mass do not suffice. The orientation of its axis also has to be given, which will involve two more numbers. In the end, one needs five numbers to specify the position of the Frisbee, and one says that all possible positions of the Frisbee constitute a five-dimensional space (or manifold). The flight of the Frisbee through the air then corresponds to a curve in this manifold. Studying the positions of even more complicated (for example, hinged) objects, one is led to even higher dimensional manifolds, and, if the object lacks any rigidity -- think of a rubber band -- to infinite-dimensional ones. The project is concerned with fundamental properties of such infinite-dimensional manifolds and of the attendant functions of infinitely many variables. A main goal is to understand how local information on these manifolds and functions can be assembled into global information. The mathematical construct that tells us to what extent this is possible is called a sheaf cohomology group, and such groups will be the main objects of the research. Various components of this work have first arisen in other parts of mathematics and in theoretical physics (quantum field theory and string theory), so the project, if successful, should have some relevance to those disciplines. However, this is going to be fundamental research, and the principal investigator does not expect immediate applications outside mathematics. On the other hand, graduate students will be involved in the research. The project will thus contribute to the training of future researchers and educators.