Algebraic Geometry Over Groups

9970618研究平面曲线及其高维类似物，如曲面，是数学家普遍感兴趣的一门高度发展的学科。这项建议的目的是继续发展一门类似的学科，称为群上的代数几何，专门适用于群论。出现了许多新的想法，使我们不仅可以发展一个类似于代数几何的一般群理论，而且还可以用一种新的方式来看待各种类型的群。特别是，有一些群在某种意义上是曲线的对应物，即所谓的单相关群，现在可以从非常不同的角度来研究它们。此外，来自计算群论的技术也发挥了作用，我们也计划在这个提议的期限内利用这些技术。对称性的概念是我们大多数人都熟悉的。例如，墙纸图案，自然产生的水晶的美丽排列，以及许多旧教堂的地板瓷砖，只是几个对称的例子。数学家们发明了一种高度抽象的物体，称为群，它可以让他们捕捉和测量对称性。对这类天体的一般研究，称为群论，就是考虑到这一点而进行的。这一理论适用于许多现代数学，也适用于物理、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。这项建议将致力于进一步发展这些群体的这一一般理论。

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.