Geometric and Algebraic Topology

本项目研究流形拓扑的各个方面：它们的分类、它们的束和它们的对称性。该项目将侧重于六个领域。首先是使用外科理论、代数k -和l-理论以及法雷尔-琼斯猜想的工具，给出拓扑等变刚性的系统方法。二是研究由Anderson和Davis定义的矩阵束的特征类，并将其应用于组合入射几何。第三个领域是研究涉及3流形自同伦等价的刚性猜想及其与高维拓扑的联系。第四个领域是给出具有透镜空间上某些环面束的总空间同伦类型的流形的分类，直至同胚。这是法雷尔-琼斯猜想的一个应用。第五部分是计算群的自由积的l群，从而解决连通和问题——当一个流形本身等价于连通和的同伦时，它是一个连通和。最后从同调群作用和Smith理论的角度研究环面上p群作用的代数拓扑和点集拓扑。几何拓扑学是对流形的研究。一个n维流形是在n维欧几里德空间上局部建模的点的集合。例如，一个2流形是一个表面，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，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，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。为了理解和分类流形，人们使用各种工具，包括代数拓扑、束理论和微分几何。

This project examines various aspects of the topology of manifolds: their classification, their bundles, and their symmetries. The project will focus on six areas. The first is to give a systematic approach to topological equivariant rigidity, using tools from surgery theory, algebraic K-and L-theory, and the Farrell-Jones Conjecture. The second is to study characteristics classes of matroid bundles (defined by Anderson and Davis) and to apply them to combinatorial incidence geometry. The third area is to investigate a rigidity conjecture involving self-homotopy equivalences of 3-manifolds and its connections with high-dimensional topology. The fourth area is to give the the classification, up to homeomorphism, of manifolds having the homotopy type of the total space of certain torus bundles over lens spaces. This is an application of the Farrell-Jones Conjecture. The fifth area is to compute the L-groups of a free product of groups, and thereby solve the connected sum problem - when is a manifold which is homotopy equivalent to a connected sum itself a connected sum. The last area is to study the algebraic and point-set topology of actions of p-groups on the torus from the point of view of homotopical group actions and Smith theory.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. To understand and classify manifold one uses a variety of tools including algebraic topology, bundle theory, and differential geometry.