Geometric Topology and Manifolds

几何拓扑和流形

基本信息

批准号：
1615056
负责人：
James Davis
金额：
$ 22.37万
依托单位：
Indiana University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1615056&HistoricalAwards=false
关键词：
Geometric Topology Manifolds

项目摘要

Award: DMS 1615056, Principal Investigator: James F. DavisMajor branches of theoretical mathematics include topology, algebra, geometry, analysis, and combinatorics. This project weaves these various threads together. The major focus is the study of manifolds, which are sets of points locally modeled on Euclidean space. Sample manifolds in dimension 2 are given by a plane, the surface of a torus, and the surface of a sphere. However manifolds exist in all dimensions, and their consideration leads to a rich variety of examples and uses a rich variety of tools. This project focuses on manifolds which are analytically simple but topologically complex. The connection between the disparate fields mentioned above means that one can use topology as a oracle, producing interesting questions, examples, and research in other areas of theoretical mathematics. Manifold theory connects with most areas of mathematics, as well as physical phenomena such as cosmology, string theory, and classical and quantum mechanics.The principal investigator proposes six projects. The first is the bordism of L2-acyclic manifolds. This connects with low-dimensional topology (knot concordance), with algebra (Hilberts 17th problem and the Witt group of function fields), with high-dimensional topology (a new form of surgery theory), and analysis (amenable groups and L2-betti numbers). The second gives a systematic approach to topological equivariant rigidity, using tools from surgery theory, stratified spaces, algebraic K-and L-theory, and the Farrell-Jones Conjecture. The third is to study and apply combinatorial characteristic classes based on oriented matroids. The fourth is the Nielsen Realization question, where progress in now possible due to the Farrell-Jones conjecture and the understanding of manifolds whose fundamental groups are infinite with torsion. The fifth is foundational work in algebraic L-theory, in particular the computation of the L-theory of the polynomial ring in n variables with integral coefficients and the group ring of a free product of groups. The sixth is to study Brieskorn manifolds produced by algebraic geometry, and, in turn, to use algebraic geometry to study these manifolds. These problems are all interrelated with each other; the basic theme being classification of manifolds, their bundles, and their symmetries.

奖项：DMS 1615056，首席研究员：James F. Davis 理论数学的主要分支包括拓扑学、代数、几何、分析和组合学。这个项目将这些不同的线索编织在一起。主要重点是流形的研究，流形是在欧几里德空间上局部建模的点集。维度 2 中的样本流形由平面、圆环面和球体表面给出。然而，流形存在于所有维度中，对它们的考虑导致了丰富多样的例子并使用了丰富多样的工具。该项目重点关注分析简单但拓扑复杂的流形。上述不同领域之间的联系意味着人们可以使用拓扑作为神谕，在理论数学的其他领域产生有趣的问题、例子和研究。流形理论与数学的大多数领域以及宇宙学、弦理论、经典力学和量子力学等物理现象都有联系。首席研究员提出了六个项目。第一个是 L2-无环流形的边数。这与低维拓扑（结索引）、代数（希尔伯特第 17 题和维特函数域群）、高维拓扑（手术理论的新形式）和分析（顺应群和 L2-betti数字）。第二部分给出了拓扑等变刚性的系统方法，使用手术理论、分层空间、代数 K 和 L 理论以及法雷尔-琼斯猜想的工具。三是基于有向拟阵的组合特征类的研究和应用。第四个是尼尔森实现问题，由于法雷尔-琼斯猜想和对基本群具有挠率无限的流形的理解，该问题现在可能取得进展。第五部分是代数 L 理论的基础工作，特别是具有积分系数的 n 变量多项式环和群的自由乘积的群环的 L 理论的计算。第六是研究代数几何产生的Brieskorn流形，进而利用代数几何来研究这些流形。这些问题都是相互关联的；基本主题是流形、流形束及其对称性的分类。