Geometric and Algebraic Topology

提案：DMS-9971607主要研究者：Karen Vogtmann 摘要：这是一个小组提案，涵盖了拓扑学和几何群论的广泛主题，与组合学，代数，概率和生物学有关。 具体而言，主要研究人员建议研究：流形的自同构群;节点和链接的空间;自由群的自同构群和外自同构群;群的Whitehead算法;系统发育树的空间;与超平面安排，Coxeter群和建筑物相关的马尔可夫链; Artin群在无穷远处的拓扑结构;以及有限群的生成集的图形。 本研究的一个统一的主题是，试图理解抽象的，代数结构，将它们与具体的几何对象，有点抽象的操作序列可以实现为一组基本的操作魔方。所讨论的几何对象通常非常简单和自然，有时可以作为其他科学领域现象的数学模型。例如，几何和代数已经导致了关于马尔可夫链的新结果，这些结果被用于许多科学和应用数学领域的模拟。此外，几何的见解往往导致发现新的关系之间的先验无关的概念。

Proposal: DMS-9971607Principal Investigator: Karen Vogtmann Abstract: This is a group proposal which covers a wide range of topics in topology and geometric group theory, with connections to combinatorics, algebra, probability, and biology. Specifically, the principal investigators propose to study: diffeomorphism groups of manifolds; spaces of knots and links; the automorphism group and outer automorphism group of a free group; Whitehead algorithms for groups; the space of phylogenetic trees; Markov chains associated to hyperplane arrangements, Coxeter groups, and buildings; the topology at infinity of Artin groups; and graphs of generating sets of finite groups. A unifying theme of this research is that of trying to understand abstract, algebraic constructions by relating them to concrete, geometric objects, somewhat as a sequence of abstract operations may be realized as a set of elementary manipulations of a Rubik's cube. The geometric objects in question are often very simple and natural, and can sometimes serve as mathematical models for phenomena in other scientific fields. For example, geometry and algebra have led to new results about Markov chains, which are used for simulations in many areas of science and applied mathematics. In addition, geometric insights often lead to the discovery of new relations between a priori unrelated concepts.