Mathematical Sciences: Topology

该项目涉及三名高级研究人员， 代数拓扑和几何拓扑的两个方面：（1）自同构 自由群的群，从同调代数的观点 和几何群论 调查人员最近 开始开发新的三维模型， 自由群的自同构，完善标准模型， 基本上是一维的。 (2)受控同变 流形和广义流形的拓扑，特别是， 研究了一类新的 最近发现的高维广义流形， 布莱恩特费里米奥和温伯格 群论，对称性的数学研究，是一个 几乎所有的硬科学中无处不在的线索，从量子 物理到数据加密 自由团体最少 受约束类型的组，所有其他组都来自这些组 通过添加特殊关系获得。 因此， 自由群的对称或自同构有一个特殊的 对群论的意义 流形是全局的 我们生活的宇宙的数学模型。 最近 弦理论等理论物理学的发展表明， 一个人不应该把注意力仅仅局限于三个 三维甚至四维的时空;更多 维度通常提供更准确的模型，即使我们不能 用我们的眼睛看他们。

This project involves three senior researchers working in two areas of algebraic and geometric topology: (1) Automorphism groups of free groups, from the viewpoint of homological algebra and geometric group theory. Here the investigators have recently begun the development of a new three-dimensional model for automorphisms of free groups, refining the standard model, which is essentially one-dimensional. (2) Controlled and equivariant topology of manifolds and generalized manifolds, in particular, the study of the properties of the new class of higher-dimensional generalized manifolds recently discovered by Bryant, Ferry, Mio, and Weinberger. Group theory, the mathematical study of symmetries, is a pervasive thread in almost all the hard sciences, from quantum physics to data encryption. Free groups are the least constrained types of groups, from which all other groups are obtained by adding special relations. Hence the study of the symmetries, or automorphisms, of free groups has a particular significance for group theory. Manifolds are the global mathematical models of the universe we live in. Recent developments in theoretical physics, such as string theory, show that one should not confine one's attention just to three dimensions or even the four dimensions of space-time; more dimensions often provide a more accurate model, even if we cannot see them with our eyes.