Algebraic and Geometric Topology

关于高维流形的外科理论分类有一系列猜测，最终在K理论和L理论中的法雷尔-琼斯猜想中达到顶峰。这些猜想与沿余维一子流形分裂流形密切相关。这个分裂问题又与代数K-理论和L-理论中的零群有关。这个项目有几个不同的方面。一个是解决连通和问题--什么时候同伦的流形等于连通和本身就是连通和？二是加强了L理论中的Farrell-Jones猜想，类似于Davis-Khan-Ranicki在K-理论中所得到的猜想。另一种是利用L无限二面体群理论的计算和L结晶学群理论中的法瑞尔-琼斯猜想的证明，对环面上的对合进行分类。也存在一些中低维问题，涉及透镜空间的自同伦等价的环面的映射和链接协调的某一方面。与几何拓扑学中的通常一样，目标是使用各种代数、几何和分析技术来寻找和计算分类不变量。几何拓扑学是对流形的研究。N维流形是在n维欧氏空间上局部模拟的一组点。例如，2-流形是一个曲面，在每个点附近看起来像一个平面。许多物理现象都是由流形表示的，因此，了解流形的整体结构以及可能存在的流形对科学和数学都是基本的。流形理论与数学的大多数领域相联系，也与宇宙学、弦理论、经典和量子力学等物理现象相联系。

There are a web of conjectures about the surgery theoretic classification of high-dimensional manifolds, culminating in the Farrell-Jones Conjectures in K- and L-theory. The conjectures are closely related to splitting manifolds along codimension one submanifolds. This splitting problem is in turn related to nil groups in algebraic K- and L-theory. This project has several different aspects. One is to solve the connected sum problem -- when is a manifold which is homotopy equivalent to a connected sum itself a connected sum? Another is to provide a strengthening of the Farrell- Jones Conjecture in L-theory, similar to that achieved by Davis-Khan- Ranicki in K-theory. Yet another is to classify involutions on a torus, using the computation of the L-theory of the infinite dihedral group and the proof of the Farrell-Jones Conjecture in L-theory for crystallographic groups, joint with Connolly. There are some middle and low dimensions problems too, involving mapping tori of self- homotopy equivalences of lens spaces and a certain aspect of link concordance.The goal, as usual in geometric topology, is to use a variety of algebraic, geometric, and analytic techniques to find and compute invariants for classification. Geometric topology is the study of manifolds. An n-dimensional manifold is a set of points locally modeled on n-dimensional Euclidean space. For instance, a 2-manifold is a surface and looks like a plane near each point. Many physical phenomenon are represented by manifolds, and as such, understanding the global structure of a manifold, and what possible manifolds exist, is fundamental to the sciences, as well as to mathematics. Manifold theory connects with most areas of mathematics, as well as with physical phenomena such as cosmology, string theory, and classical and quantum mechanics.