The (de Rham) cohomology, based on differential forms on a smooth manifold, provides an analytic representation of singular cohomology. For natural spaces which are not smooth, like complex algebraic varieties, singular cohomology is available, but some of its useful properties no longer hold and there is no good analytic representation. However, the de Rham cohomology of forms with moderate growth near the singular set provides a good alternative, because it satisfies duality and (Hodge) filtration properties like those in the smooth case. The investigator and his collaborators study this cohomology in two cases, where the variety has isolated, point singularities and where it is the minimal compactification of a locally symmetric space. In the latter case certain natural functions are studied, with the goal of proving that they are modular forms. A second line of investigation concerns quadratic forms defined over the ring of regular functions on an algebraic variety. The investigator relates these quadratic forms to quadratic forms defined on the rational function fields of all subvarieties, in much the same way that holomorphic functions are related to meromorphic functions and their residues. The consequent connection to certain multiplicative structures on ideals in the ring of regular functions is explored. The first part of the project pursues a program to study wrinkles (singularities) in a space. If this space were a surface, wrinkles would typically occur near places where the degree of turning (called curvature) is very high. One part of the project is then to quantify this degree of turning, with the eventual goal of understanding what sorts of wrinkles can occur singly or in groups. Some of the problems in this project come up in physics, especially string theory, to which the the investigator and his collaborators intend to apply the methods they develop. The second part of the project concerns questions about abstract number systems, in particular whether a given (abstract) number is a perfect square, or the sum of two or more perfect squares. Such questions have been studied for centuries in the mathematical subfield called number theory. More recently, the intractability of the problem of finding certain perfect squares, even with a computer, has been shown by other investigators to be the basis of certain protocols which guarantee the secure and fair electronic exchange of information.
