Quadratic Forms on Schemes and Geometry of Varieties

基于光滑流形上的微分形式的(De Rham)上同调给出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，但它的一些有用性质不再成立，也没有良好的解析表示。然而，在奇异集附近具有适度增长的形式的de Rham上同调提供了一个很好的选择，因为它像光滑情形一样满足对偶性和(Hodge)滤子性质。研究人员和他的合作者在两种情况下研究了这种上同调，在这两种情况下，簇具有孤立的点奇点，并且它是局部对称空间的最小紧化。在后一种情况下，研究某些自然函数，目的是证明它们是模形式。第二个研究方向是定义在代数簇上的正则函数环上的二次型。研究人员将这些二次型与定义在所有子簇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留数的关系一样。研究了正则函数环上某些理想乘法结构之间的必然联系。该项目的第一部分是一个研究空间中皱纹(奇点)的项目。如果该空间是一个曲面，皱纹通常会出现在转折度(称为曲率)非常高的位置附近。该项目的一部分是量化这种转折程度，最终目标是了解哪种皱纹可以单独出现或成群出现。这个项目中的一些问题出现在物理学中，特别是弦理论，研究人员和他的合作者打算将他们开发的方法应用到这个问题上。该项目的第二部分涉及关于抽象数系统的问题，特别是给定的(抽象)数是否是完美平方，或者是两个或更多完美平方的和。这类问题在被称为数论的数学分支领域已经研究了几个世纪。最近，其他研究人员已经证明，即使用计算机也难以找到某些完美正方形的问题，这是某些协议的基础，这些协议保证了信息的安全和公平的电子交换。

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.