Algebraic Geometry Over Groups

群上的代数几何

• 批准号: 9970618
• 负责人: Gilbert Baumslag
• 金额: --
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970618&HistoricalAwards=false
• 关键词: Algebraic; Geometry; Over; Groups

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.

9970618 平面曲线及其高维类似物（例如曲面）的研究是数学家普遍感兴趣的高度发达的学科。该提案的目的是继续发展一个类似的学科，称为群上的代数几何，它特别适用于群论。许多新思想的出现，使我们不仅能够发展出类似于代数几何中的群的一般理论，而且能够以新的方式看待各种类型的群。特别是有一些群在某种意义上与曲线相对应，即所谓的单相关群，现在可以从一个非常不同的角度来研究它们。此外，计算群论的技术也开始发挥作用，我们也计划在本提案的任期内利用它们。对称性的概念对我们大多数人来说都很熟悉。例如，壁纸图案、天然水晶的美丽排列以及许多古老教堂的地砖只是对称的几个例子。数学家发明了一种高度抽象的对象，称为群，它使他们能够捕获和测量对称性。对这些物体的一般研究，称为群论，就是在考虑到这一点的情况下进行的。该理论适用于许多现代数学以及物理、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。本提案将致力于进一步发展这些群体的一般理论。