Geometric Topology and Manifolds

奖：DMS 1615056，首席研究员：詹姆斯·F·戴维斯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编织在一起。主要的焦点是流形的研究，流形是在欧几里德空间上局部模拟的点集。2维的样本流形由平面、环面和球面给出。然而，流形存在于所有维度中，它们的考虑导致了丰富的各种例子，并使用了丰富的各种工具。这个项目的重点是分析简单但拓扑复杂的流形。上面提到的不同领域之间的联系意味着人们可以使用拓扑学作为一种预言，产生有趣的问题、例子和理论数学的其他领域的研究。流形理论与数学的大多数领域，以及宇宙学、弦理论、经典和量子力学等物理现象相联系。首席研究员提出了六个项目。第一个是L2-非循环流形的边论。这与低维拓扑(纽结协调性)、代数(Hilberts第17问题和函数域的Witt群)、高维拓扑(一种新形式的外科理论)和分析(服从群和L2-Betti数)相联系。第二章利用外科理论、分层空间、代数K-理论和L理论以及Farrell-Jones猜想等工具，系统地研究了拓扑等变刚性。三是基于定向拟阵的组合特征类的研究与应用。第四个是Nielsen实现问题，由于Farrell-Jones猜想和对其基本群为无穷扭转的流形的理解，现在取得进展是可能的。第五部分是代数L理论的基础性工作，特别是关于整系数n元多项式环和群的自由积的群环的计算。第六是研究由代数几何产生的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、流形丛和流形对称的分类。

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.