Quadratic Forms on Schemes and Geometry of Varieties

基于光滑流形上的微分形式的（de Rham）上同调，提供了奇异上同调的解析表示。对于不光滑的自然空间，如复代数簇，奇异上同调是可用的，但它的一些有用的性质不再成立，也没有好的解析表示。然而，在奇异集附近有适度增长的形式的de Rham上同调提供了一个很好的选择，因为它满足对偶性和（霍奇）过滤性质，就像在光滑情况下一样。调查员和他的合作者研究这种上同调在两种情况下，其中品种孤立，点奇点，并在那里它是最小的紧化的局部对称空间。在后一种情况下，研究某些自然函数，目的是证明它们是模形式。第二条线的调查关注二次形式定义的环上的正规职能的代数品种。调查涉及这些二次形式的二次形式定义的有理函数领域的所有子品种，在很大程度上相同的方式，全纯函数有关的亚纯函数及其残留物。因此，连接到某些乘法结构的理想环的正规函数进行了探讨。 该项目的第一部分是研究空间中的褶皱（奇点）。如果这个空间是一个曲面，褶皱通常会出现在转向度（称为曲率）非常高的地方附近。该项目的一部分是量化这种转向程度，最终目标是了解单独或成组出现的皱纹类型。这个项目中的一些问题出现在物理学中，特别是弦理论，研究者和他的合作者打算应用他们开发的方法。该项目的第二部分涉及有关抽象数字系统的问题，特别是一个给定的（抽象）数字是否是一个完美的平方，或两个或多个完美平方的总和。这类问题在数论这一数学分支中已经研究了几个世纪。最近，即使用计算机也难以找到某些完美的正方形，这一问题已被其他研究人员证明是某些保证安全和公平的电子信息交换协议的基础。

The (de Rham) cohomology, based on differential forms on a smoothmanifold, provides an analytic representation of singular cohomology. For natural spaces which are not smooth, like complex algebraicvarieties, 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.