Mathematical Sciences: Moduli Problems in Algebraic Geometry

9500964撒迪厄斯教授撒迪厄斯将使用他的翻转技术研究商簇的拓扑和几何。特别是，他将研究某些射影空间的配置空间。他还将研究向量丛和曲线上的模空间。他希望计算上同调环和相关的希尔伯特多项式中的关系。这是代数几何领域的研究，但它与本世纪理论物理的两大进步--量子力学和广义相对论--直接相关。代数几何本身是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。***

9500964 Thaddeus Professor Thaddeus will study the topology and geometry of quotient varieties using his techniques of flips. In particular he will study configuration spaces of certain projective spaces. He will also study moduli spaces off vector bundles and curves. he hopes to compute relations in their cohomology rings and the associated Hilbert polynomials. This is research in the field of algebraic geometry, yet it directly connects to two of the great advances in theoretical physics in this century--quantum mechanics and general relativity. Algebraic geometry itself is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics. ***