Mathematical Sciences: The Structure of Algebraic Varieties

9622394首席研究员的目的是研究原木品种。它们可以被认为是由一个变量和一个除数组成的一对。这些对象出现在一些问题的研究中，例如最小模型理论、模问题和合理连接变量的两族分类。对于对数变量，可以将各种数值不变量（例如差异数和自交数）联系起来，这些不变量表现出非常有趣的行为。对于普通变量，这些不变量是整数，但对于对数变量，它们是有理数。在许多情况下，并非所有有理数都以这种方式出现，理解出现的有理数集合是至关重要的。即使在二维空间中，这也是一个很难回答的问题，而这个问题的答案对于曲面的模（Alexeev, Koll\ ar,Mori）、Fano纤维空间的双元变换（Corti）和投芬图几何（Hassett）有着直接而深刻的应用。首席研究员还旨在进一步探索这些最新应用以及相关思想在高性能计算理论方面的应用（与Babai等人合作）。首席研究员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。在它的起源中，它处理的图形可以用最简单的方程在平面上定义，比如直线、圆、椭圆。如今，该领域不仅使用代数和几何的方法，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。

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.