Algebraic Geometry Over Groups

9970618对平面曲线及其高维类似物（如曲面）的研究，是数学家普遍感兴趣的一门高度发展的学科。本建议的目的是继续发展一个类似的主题，称为群上的代数几何，它特别适用于群论。出现了许多新的思想，这些思想不仅使我们能够发展出一种关于群的一般理论，类似于代数几何中的理论，而且还使我们能够以一种新的方式来看待各种各样的群。特别是有一些群在某种意义上是曲线的对应物，所谓的单相关群，现在可以从一个非常不同的有利位置来研究。此外，计算群论的技术也会发挥作用，我们计划在本提案的任期内也利用它们。对称的概念对我们大多数人来说都很熟悉。例如，墙纸的图案、自然形成的水晶的美丽排列以及许多古老教堂的地砖都是对称的几个例子。数学家们发明了一种高度抽象的对象，称为群，这使他们能够捕捉和测量对称性。对这类物体的一般研究，称为群论，就是在这样的思想下进行的。这一理论在现代数学、物理、化学、生物学、计算理论、密码学以及我们的三维和四维物理世界中都有应用。本建议将致力于进一步发展这种关于这些群体的一般理论。

9970618 The study of plane curves and their higher-dimensional analogues such as surfaces, is a highly developed subject of great general interest to mathematicians. The objective of this proposal is to continue with the development of an analogous subject, termed algebraic geometry over groups, which applies specifically to group theory. A number of new ideas have arisen which allow us to not only develop a general theory for groups, akin to that in algebraic geometry, but also to look at various classes of groups in a new way. In particular there are groups which in a sense are the counterpart to curves, so-called one-relator groups, which can now be studied from a very different vantage point. In addition, techniques from computational group theory come into play as well and we plan on exploiting them also, in the tenure of this proposal.The notion of symmetry is familiar to most of us. For example, wall-paper patterns, the beautiful arrangements of naturally occurring chrystals and the floor tilings of many old churches are but a few examples of symmetry. Mathematicians have invented a highly abstract object, called a group, which allows them to capture and measure symmetry. The general investigation of such objects, called group theory, has been carried out with this in mind. This theory has applications to much of modern mathematics as well as to physics, chemistry, biology, to the theory of computing, to cryptography as well as to our physical world of three and four dimensions. It is to the further development of this general theory of these groups that this proposal will be devoted.