Algebraic and Geometric Topology

关于高维流形的外科手术理论分类，有一个网络，最终在K-和L-理论中的法雷尔-琼斯猜想。 这些拓扑与沿沿着余维子流形分裂流形密切相关。 这个分裂问题又与代数K-和L-理论中的诣零群有关。 这个项目有几个不同的方面。 一个是解决连通和问题--什么时候一个同伦等价于一个连通和的流形本身是一个连通和？ 另一个是在L-理论中加强Farrell- Jones猜想，类似于Davis-Khan- Ranicki在K-理论中实现的。 另一个是对环面上的对合进行分类，使用无限二面体群的L-理论的计算和结晶群的L-理论中的法雷尔-琼斯猜想的证明，与康诺利联合。 也有一些中、低维的问题，涉及到透镜空间的自同伦等价的映射环面和链接协调的某个方面。目标和几何拓扑学中一样，是使用各种代数、几何和分析技术来寻找和计算分类的不变量。几何拓扑学是研究流形的学科。n维流形是在n维欧氏空间上局部建模的一组点。例如，2-流形是一个曲面，在每个点附近看起来像一个平面。许多物理现象都是用流形来表示的，因此，理解流形的整体结构以及可能存在的流形是科学和数学的基础。 流形理论与数学的大多数领域都有联系，也与宇宙学、弦理论、经典力学和量子力学等物理现象有联系。

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.