Quadratic Forms on Schemes and Geometry of Varieties

基于光滑流形上的微分形式，给出了奇异上同调的解析表示。对于非光滑的自然空间，如复代数变异，奇异上同是可用的，但它的一些有用性质不再成立，并且没有很好的解析表示。然而，在奇异集附近适度增长的形式的de Rham上同调提供了一个很好的选择，因为它满足了光滑情况下的对偶性和（Hodge）过滤性质。研究者和他的合作者在两种情况下研究了这种上同调，其中变种具有孤立的点奇点，并且它是局部对称空间的最小紧化。在后一种情况下，我们研究了某些自然函数，目的是证明它们是模形式。第二项研究涉及在代数变量上正则函数环上定义的二次型。研究者将这些二次型与定义在所有子变种的有理函数域上的二次型联系起来，就像全纯函数与亚纯函数及其残基联系起来一样。探讨了正则函数环上理想的若干乘法结构的后续联系。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表面，皱纹通常会发生在旋转程度（称为曲率）非常高的地方附近。该项目的一部分是量化这种旋转的程度，最终目标是了解哪些类型的皱纹可以单独或成群发生。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将他们开发的方法应用于弦理论。项目的第二部分涉及抽象数字系统的问题，特别是给定的（抽象）数字是完全平方数，还是两个或多个完全平方数的和。这些问题已经在数论的数学分支领域研究了几个世纪。最近，即使使用计算机也很难找到某些完全平方的问题，这一难题已被其他研究人员证明是某些协议的基础，这些协议保证了安全、公平的电子信息交换。

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.