Geometric and Algebraic Topology

摘要：这是一个小组提案，涵盖了拓扑和几何群论的广泛主题，并与组合学、代数、概率论和生物学有联系。具体来说，主要研究者建议研究：流形的微分同构群；结和链的空间；自由群的自同构群和外自同构群；群体的Whitehead算法；系统发育树的空间；与超平面排列、Coxeter群和结构相关的马尔可夫链；Artin群无穷远处的拓扑；和有限群的生成集的图。本研究的一个统一主题是，通过将抽象的代数结构与具体的几何对象联系起来，试图理解抽象的代数结构，就像一系列抽象操作可以被实现为一组基本的魔方操作。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，几何和代数导致了关于马尔可夫链的新结果，它被用于许多科学和应用数学领域的模拟。此外，几何洞见常常导致发现先验的不相关概念之间的新关系。

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.