Geometric Topology and Manifolds

奖项：DMS 1615056，主要研究者：James F.戴维斯理论数学的主要分支包括拓扑学、代数学、几何学、分析学和组合学。 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，流形是在欧几里得空间上局部建模的点的集合。 二维流形的样本由平面、环面的表面和球面给出。 然而流形存在于所有维度中，对它们的考虑导致了丰富多样的例子，并使用了丰富多样的工具。 这个项目的重点是流形，这是分析简单，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使用拓扑学作为神谕，在理论数学的其他领域产生有趣的问题，例子和研究。流形理论与大多数数学领域以及宇宙学、弦理论、经典力学和量子力学等物理现象都有联系。 第一个是L2-无圈流形的边数。 这与低维拓扑（结一致性），代数（希尔伯特第17问题和函数场的维特群），高维拓扑（一种新形式的外科手术理论）和分析（顺从群和L2-贝蒂数）有关。第二个给出了一个系统的方法拓扑等变刚性，使用工具从外科手术理论，分层空间，代数K-和L-理论，和法雷尔-琼斯猜想。 三是研究和应用基于有向拟阵的组合特征类。 第四个是尼尔森实现问题，由于Farrell-Jones猜想和对基本群具有无穷挠的流形的理解，现在可能取得进展。 第五是基础性的工作，代数L-理论，特别是计算的L-理论的多项式环在n个变量的整数系数和群环的自由产品的群体。 第六是研究由代数几何产生的Brieskorn流形，反过来，利用代数几何来研究这些流形。 这些问题都是相互关联的;基本的主题是分类的流形，他们的丛，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.