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.
This project was about complex analysis and geometry, often in connection with infinite dimensional spaces. In a nutshell, this means the study of fundamental properties of functions of complex variables and of the geometric figures they describe; infinite dimensionality refers to functions that have infinitely many dependent or independent variables (or both). While this research is in pure mathematics, some of its aspects are motivated by physics, where functions of compex variables, even infinitely many, often occur in the desciption of fields (electromagnetic or more complicated fields) and of other objects. Through the physical connection the research has the potential to contribute to our understanding of the microworld, whose quantum physical description depends on complex variables and functions of these complex variables. One line in this research took up a problem in physics. When a physical system is described quantum mechanically, the principles of quantization do not unambiguously determine this description, but usually the descriptions depend on a few choices. The question is whether different choices produce equivalent descriptions and give the same predictions for the behavior of the system. Following earlier work by physicists, with a collaborator we constructed a framework in which the equivalence of descriptions can be related to a certain quantity, the ``curvature", being zero. Using this we proved that for certain systems quantization is indeed independent of choices, while for others it is not. Another line in my research concerns metrics on complex manifolds. A gross simplification of the motivating problem will help understand the origin of this line. Imagine a rubber band, that we are free to arrange in some shape in the plane or in our 3 dimensional space: a circle, an ellipse, a rectangle, or some other irregular shape. Which is the optimal shape? (It is ``shape" that corresponds to the notion of ``metric" above.) Even without precisely defining what we mean by optimal, we can argue that a circular shape is the best. For in a circular shape, if one knows the length of the rubber band, all other geoemtric quantities can be immediately derived: the area enclosed, the greatest distance among the points on the band, its curvature, etc. Nothing like this would be true for other shapes.---Without simplifications, then, the mathematical question is, given a complex manifold (that is, a geometrical figure that generalizes the rubber band of the example), which metric on it (i.e., which shape) has the best properties? To investigate and answer this problem one should first understand the properties of all metrics on our complex manifold. With a collaborator we solved a problem that had been around for some 25 years on the totality (``the space") of all these metrics. We showed that this infinite dimensional space of metrics is very different from familiar finite dimensional spaces, in that two metrics within this space cannot necessarily be connected by a curve that is shorter than all other connecting curves.This is in contrast with the familiar fact that given two points in our 3 dimensional space, there is a shortest among all curves connesting the two, namely the straight line segment. Most other research done with support of this grant clarified to what extent functions of infinitely many variables and the corresponding infinite dimensional figures have properties similar to functions of finitely many variables, and how they are different. The research involved three of my PH. D. students.
