Problems In Algebraic Geometry

该奖项支持在代数几何的一些问题上的工作。首先是探索高维复杂几何中的不变量与丰富的经典几何中的阿贝尔变分和黎曼曲面之间的联系。该负责人还建议继续他在不规则变量的几何上的工作，以及在任意维的光滑投影变量的非常一般的点上寻找充足线束的局部正性的下界的问题。最后，他将研究最近与Bo Ilic一起观察到的一些问题，即如果X是具有非负余切束的n维的射影变化，那么X的射影嵌入的最小度必须在n中本质上呈指数增长。这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

Lazarsfeld 9713149 This award supports work on a number of problems in algebraic geometry. The first is to explore the connections between invariants from higher dimensional complex geometry and the rich classical geometry of abelian varieties and Riemann surfaces. The principal investigator also proposes to continue his work on the geometry of irregular varieties, as well as on the problem of trying to find lower bounds on the local positivity of an ample line bundle at a very general point of a smooth projective variety of arbitrary dimension. Finally, he will investigate some questions growing out of a recent observation with Bo Ilic to the effect that if X is a projective variety of dimension n with non-negative cotangent bundle, then the least degree of a projective embedding of X must grow essentially exponentially in n. This is research in the field of algebraic geometry. Algebraic geometry 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.