Geometric and Algebraic Topology

这个项目检查流形拓扑的各个方面：它们的分类，它们的丛和它们的对称性。 该项目将侧重于六个领域。 首先是给出一个系统的方法拓扑等变刚性，使用手术理论，代数K-和L-理论，和法雷尔-琼斯猜想的工具。 第二部分是研究拟阵丛的特征类（由安德森和戴维斯定义），并将其应用于组合关联几何。 第三个领域是研究一个刚性猜想，涉及3-流形的自同伦等价及其与高维拓扑的联系。 第四个领域是给的分类，同胚，流形的同伦型的总空间的某些环面丛在透镜空间。这是Farrell-Jones猜想的一个应用。 第五个领域是计算群的自由积的L-群，从而解决连通和问题--当一个流形同伦等价于一个连通和时，它本身就是一个连通和。 最后一个领域是从同伦群作用和Smith理论的角度来研究环面上p-群作用的代数和点集拓扑。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。 为了理解和分类流形，人们使用了各种各样的工具，包括代数拓扑，丛理论和微分几何。

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.