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Geometry and Groups: Enumeration and Finite Representations

几何和群：枚举和有限表示

基本信息

批准号：
1812153
负责人：
David McReynolds
金额：
$ 22.1万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812153&HistoricalAwards=false
关键词：
Geometry Groups Enumeration Finite Representations

项目摘要

The richest symmetry groups in nature are exemplified by the collection of all rotations of a two-dimensional sphere or the collection of all rigid motions of the Euclidean plan. The operations of applying two symmetries in order, and of inverting a symmetry to get another one, endow these sets of symmetries with the structure known in mathematics as a group. Many other groups arise as subgroups of these groups of continuous motions, modeled on the group of rigid motions of a planar tiling and its relationship to the full group of isometries of the Euclidean plan. Number theory give rise to many of these examples, with constructions that are analogous to the discrete, widely separated way that the integers sit within the real number line. One of the projects to be pursued seeks to find a computationally feasible way to list all of the arithmetically defined discrete subgroups of motion within the most important groups of continuous motions.The continuous groups of motion referred to above are known in mathematics as semisimple Lie groups; Sophus Lie was the nineteenth-century mathematician who discovered the basic structural facts of such groups of continuous transformations, and semisimplicity is an algebraic property shared by most of the important examples. The first line of work planned will recursively enumerate arithmetic lattices in semisimple Lie groups, providing a solution to the isomorphism problem for this class of groups and addressing a conjecture of Belolipetsky and Lubotzky on the number of isometries between distinct finite covers of an arithmetic manifold. A second line of investigation concerns the relationship between a manifold and its finite-sheeted covering spaces, which is encoded in the profinite completion of the manifold's fundamental group.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

自然界中最丰富的对称群的例子是二维球体的所有旋转的集合或欧几里得平面的所有刚性运动的集合。将两个对称性按顺序应用，以及将一个对称性反转以得到另一个对称性的操作，赋予了这些对称性集合以数学中已知的群结构。许多其他的群作为这些连续运动群的子群出现，以平面平铺的刚性运动群及其与欧几里得平面的等距全群的关系为模型。数论给出了许多这样的例子，其结构类似于离散的、广泛分离的整数位于真实的数行中的方式。其中一个项目是寻找一种计算上可行的方法，列出最重要的连续运动群中所有算术定义的离散运动子群。索菲斯·李是19世纪的数学家，他发现了这种连续变换群的基本结构事实，而半单性是大多数重要例子所共有的代数性质。第一行的工作计划将递归枚举算术格在半单李群，提供了一个解决方案的同构问题，这类群体和解决一个猜想Belolipetsky和Lubotzky之间的数量isometries不同的有限覆盖的算术流形。第二条调查线涉及流形和它的有限片覆盖空间之间的关系，这是编码在流形的基本group.This奖项的profinite完成反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。