Geometric Topology and Manifolds

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，代数，几何，分析和组合。这个项目将这些不同的线索编织在一起。主要的焦点是流形的研究，流形是在欧几里德空间上局部建模的点的集合。二维流形的样例由平面、环面和球面给出。然而，流形存在于所有维度中，对它们的考虑导致了各种各样的例子，并使用了各种各样的工具。这个项目的重点是流形的分析简单，但拓扑复杂。上面提到的不同领域之间的联系意味着人们可以将拓扑用作预言器，在理论数学的其他领域产生有趣的问题、示例和研究。流形理论与数学的大多数领域，以及物理现象，如宇宙学、弦理论、经典力学和量子力学联系在一起。首席研究员提出了六个项目。第一个是l2 -无环流形的本体。这与低维拓扑（结调和）、代数（Hilberts第17问题和函数场的Witt群）、高维拓扑（外科理论的一种新形式）和分析（可服从群和L2-betti数）有关。第二部分给出了拓扑等变刚性的系统方法，使用来自外科理论、分层空间、代数k和l理论以及法雷尔-琼斯猜想的工具。三是研究和应用基于定向拟阵的组合特征类。第四个是尼尔森实现问题，由于法雷尔-琼斯猜想和对其基本群具有无限挠性的流形的理解，现在可能取得进展。第五部分是代数l论的基础工作，特别是n变量整系数多项式环的l论计算和群的自由积群环的l论计算。六是研究由代数几何产生的布里斯科恩流形，进而用代数几何来研究这些流形。这些问题都是相互关联的；基本的主题是流形的分类，他们的束，和他们的对称性。

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.