Mathematical Sciences: The Structure of Algebraic Varieties

9622394 主要研究者的目标是研究原木品种。这些可以被认为是一对组成的品种和一个除数。这些对象出现在几个问题的研究，例如最小模型理论，模问题和双有理分类的合理连接品种。 对于一个对数簇，人们可以将各种数值不变量（例如差异和自相交数）联系起来，这些不变量表现出非常有趣的行为。对于普通的变种，这些不变量是整数，但对于对数变种，它们是有理数。在许多情况下，并不是所有的有理数都是这样产生的，理解产生的有理数集是至关重要的。即使在二维这是一个很难的问题，答案有直接和深刻的应用模的表面（Alexeev，Koll'ar，森），双有理变换的法诺纤维空间（Corti）和丢番图几何（Hassett）。 首席研究员还旨在进一步探索这些和相关思想在高性能计算理论方面的最新应用（与巴拜等人的联合工作）。 主要研究者的目标是在代数几何领域进行研究。代数几何是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，它处理的图形，可以定义在平面上的最简单的方程，如直线，圆，椭圆。如今，该领域不仅使用代数和几何的方法，而且还使用分析和拓扑学的方法，相反，这些方法在这些领域以及物理学，理论计算机科学和机器人学中也得到了应用。

9622394 The principal investigator aims to study log varieties. These can be thought of as a pair consisting of a variety and a divisor on it. These objects arise in the study of several questions, for instance minimal model theory, moduli problems and the birational classification of rationally connected varieties. To a log variety one can associate various numerical invariants (for instance discrepancies and selfintersection numbers), and these invariants exhibit very interesting behaviour. For ordinary varieties these invariants are integers, but for log varieties they are rational numbers. In many cases not all rational numbers arise this way, and it is crucial to understand the set of rational numbers that arise. Even in dimension two this is a hard question,and the answer has immediate and deep applications to the moduli of surfaces (Alexeev, Koll\'ar,Mori), birational trasformations of Fano fiber spaces (Corti) and to diophantine geometry (Hassett). The principal investigator also aims to further explore recent applications of these and related ideas to the theoretical aspects of high-performance computing (joint work with Babai et al). The principal investigator aims to conduct research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of 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, like lines, circles, ellipses. Nowadays the field makes use of methods not only from algebra and geometry, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.