Complex Analysis and Geometry

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The principal investigator proposes to work on problems of complex analysis and geometry that involve infinite dimensional objects. The project has three main themes. The first is to study the sheaf cohomology groups of certain infinite-dimensional manifolds and their loop spaces. The huge amount of experience from finite dimensions and the much smaller, but growing, amount of experience in infinite dimensions suggest that cohomological questions will be central to analysis and geometry in infinite dimensions. The overarching questions are: (1) How is the infinite-dimensional theory different from the finite-dimensional theory, and how are the two similar? (2) What kind of information concerning a manifold can be gleaned from the cohomology groups of its loop spaces? A concrete problem to be considered is to endow the cohomology groups of infinite-dimensional manifolds with a suitable topology and to ask whether theorems like Leray's isomorphism theorem or a relative version of Serre duality theorem can be proved. The second theme of the project concerns direct images of Hermitian holomorphic vector bundles (over finite-dimensional manifolds) under not-necessarily-proper maps. The best one can expect is that direct images will be smooth fields of Hilbert spaces, a recent generalization of vector bundles. The principal investigator will explore topics such as the following ones: geometric conditions that imply the direct image is indeed a smooth field of Hilbert spaces; the curvature of the direct image; and the theory of the Cauchy-Riemann equations in holomorphic fields of Hilbert spaces. The third theme concerns the space of Kahler metrics on a given Kahler manifold. This is an infinite-dimensional Riemannian manifold, and the project will investigate its geodesics.One of the roles of mathematics is to provide the terms in which to describe the world around us. As we are discovering more and more complicated phenomena, both natural and societal, it is critical that the descriptions nevertheless stay simple. Mathematics achieves this by introducing new notions. Here is an example that is pertinent to this project. Quantities in the real world are measured by real numbers, and since the invention of analytic geometry we know that real world figures (e.g., curves, surfaces) can be described by functions of real variables. Yet real world quantities in oscillatory phenomena, for example voltage in alternating current, can be described much more simply in terms of complex numbers. It takes some investment of time and energy to introduce complex numbers, but once this is done, the descriptions of the phenomena become fully transparent. By now we understand well that complex numbers are indispensable in a huge number of problems. Similarly, one is often forced to pass from functions of real variables and the figures they describe to functions of complex variables and the associated geometric figures. Real world phenomena often involve a vast number of parameters to control, and their treatment requires working with functions of a vast number of variables. It again turns out that introducing a new mental construct, functions of infinitely many variables, oftentimes simplifies the understanding of those phenomena. This project, continuing earlier research of the prinicipal investigator, is directed towards uncovering fundamental properties of functions of infinitely many complex variables and the associated figures, so-called infinite-dimensional complex manifolds. Part of the project will investigate what bearing the infinite-dimensional theory has on finite-dimensional, closer-to-home problems. Some of the concrete problems in the project are directly motivated by quantum theory and have the potential to impact theoretical physics. By involving graduate students, the project will also serve to introduce young people to mathematical research.