Relative De Rham Complexes, Families of Varieties

Kleiman Kovacs 9600089该奖项支持S.Kovacs博士作为S.Kleiman教授的博士后助理致力于复杂代数几何问题的研究。他希望为非光滑态射的相对De Rham复形找到一种构造。他还希望研究复变数的对数反转，并试图证实Kollar的猜想，即某些在单点爆炸的阿贝尔曲面会产生锥体，这些锥体分解为非平凡的有限生成部分和非平凡的无处局部有限生成部分。这项研究是在代数几何领域进行的，代数几何是现代数学中最古老的部分之一，但在过去的10年里，它发展到了解决了几个世纪以来的问题的地步。最初，它用最简单的方程，即多项式来处理平面上定义的图形。今天，这个领域不仅使用代数的方法，而且还使用分析和拓扑学的方法，相反，它在这些领域被广泛使用。此外，它在物理、理论计算机科学、密码学、编码理论和机器人学等领域都证明了自己的有用。

Kleiman Kovacs 9600089 This awards supports the research of Dr. S. Kovacs as a post-doctoral associate of Professor S. Kleiman to work on problems in complex algebraic geometry. He hopes to find a construction for the relative De Rham complex for non-smooth morphisms. He also hopes to study log flips of complex varieties and to try to confirm Kollar's conjecture that certain Abelian surfaces blown-up at a single point give cones which decompose into non-trivial finitely generated part and a non-trivial nowhere locally finitely generated part. This research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which blossomed to the point where it has, in the past 10 years, solved problems that have stood for centuries. Originally, it treated figures defined in the plane by the simplest of equations, namely polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.