Mathematical Sciences: Fundamental Groups of Complex Manifolds

该奖项支持J. Carlson教授和D. Toledo教授在代数几何方面的研究。他们将研究复杂流形的基本群及其表示。他们打算研究最近构造的具有非剩余有限基本群的流形实例的表示理论。他们还打算应用调和映射理论来扩大已知Shafarevic猜想为真的群的类别。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，它已经发展到解决了几个世纪以来一直存在的问题的程度。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，该领域不仅使用代数的方法，而且还使用分析和拓扑的方法，相反，它在这些领域中被广泛使用。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人等多种领域都很有用。

This awards supports the research of Professors J. Carlson and D. Toledo to work in algebraic geometry. They will work on fundamental groups of complex manifolds and their representations. They intend to study the representation theory of the recently constructed examples of manifolds with non-residually finite fundamental groups. They also intend to apply the theory of harmonic maps to enlarge the class of groups for which the Shafarevic conjecture is known to be true. The 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.