Global motivic integration and the cohomology of moduli spaces

最简单地说，代数几何是研究多变量多项式方程的解的几何。为了继续进行，必须将解空间的集合扩大到更大的集合，该集合由作为此类空间的代数粘合的空间组成。我们将把这样的空间称为变化。在20世纪50年代，格罗森迪克学派的S告诉我们，记住定义方程也很重要。这种解空间的几何性质通常是通过将一个代数不变量与该空间联系起来来研究的。1995年，在微积分和数论思想的启发下，M.Kontsevich证明了与一类特殊变元有关的一些不变量的猜想。动机整合理论应运而生。这一理论是在过去十年中发展起来的，最引人注目的是J.Denef和J.F.Loeser。本研究的目的是进一步扩展这一理论。这个建议的第二部分旨在研究代数曲线上一种特殊的簇，即所谓的半稳定向量丛的模空间的一些不变量。

Algebraic geometry, at the crudest level, is the study of the geometry of solutions of polynomial equations in many variables. In order to proceed one must enlarge the collection of solution spaces to a larger collection consisting of spaces that are algebraic gluings of such spaces. We will refer to such spaces as varieties. In the 1950's the Grothendieck school taught us that it is also important to remember the defining equations. The geometry of such solution spaces is usually studied by associating to the space an algebraic invariant. In 1995, inspired by ideas from calculus and number theory, M. Kontsevich proved some conjectures pertaining to some invariants associated to a special class of varieities. The theory of motivic integration was born. The theory has been developed over the last ten years most notably by J. Denef and  F. Loeser. The aim of this research is to further extend this theory. The second part of this proposal aims to study some invariants of the a special variety, the so called moduli space of semistable vector bundles over an algebraic curve.