Graphs, Trees and Geometric Group Theory

DMS-0204185 Karen Vogtman这个项目继续发展一套想法，从试图通过将自由群表示为图的同伦等价来理解自由群的自同构群开始。所有这种同伦等价的空间称为外空间，是一个可压缩空间，在这个空间上，自由群的外自同构群适当地不连续地作用。通过这一行动分析外层空间的商，多年来已经产生了关于这一群体的大量代数信息。最近，Maxim Kontsevich的工作将这个商的有理同调与某个无限维李代数的同调联系起来，该李代数由自由李代数的导子组成。Kontsevich还定义了这种无限维李代数的变分，它的同调与映射类群的同调有关，而ToKontsevich的图同调是一种包含各种三维流形和纽结不变量的同调理论。首席研究者将利用这一观点来获得拓扑学家感兴趣的这些经典群的不变量的新信息；相反，将应用拓扑方法来获得关于这些李代数及其同调的新信息。特别地，链复形上用于计算图同调的新的李双代数结构将被研究，这些结构是由首席投资人Jim Conant最近发现的。最后，对外层空间的局部几何进行了研究。外层空间中的某些邻域是非正弯曲的度量空间，并且可以用有限标号度量树空间来标识。这种树在包括生物学在内的许多科学分支中都很有趣，在生物学中，它们看起来像是由DNA数据产生的进化树。这个项目的另一个目标是确定在这个空间中寻找测地线的有效算法，并从那里继续开发一个将统计方法应用到这个树木空间的程序。在科学和数学中，图形和树出现在许多环境中。在几何群论中，一种称为外空间的图空间被广泛地用于研究有限生成自由群的外自同构群。该空间最初是由Marc Culler和首席研究者定义的。在这个项目中，首席研究员将继续研究这个空间的几何与自同构群的代数性质之间的关系。康采维奇的工作为这个问题提供了一个新的视角；出于物理学和辛几何的考虑，康采维奇定义了各种抽象的无限维代数结构，这些结构的不变量与外层空间的不变量密切相关。最后，外层空间的小区域可以用有限树空间来识别。这种树被进化生物学家用作工具，这个项目的另一个重点是以非常具体的算法方式理解这些区域的局部几何图形，这可以应用于分子系统发生学中的问题。

DMS-0204185Karen VogtmanThis project continues to develop a set of ideas which began with an attempt to understand automorphism groups of free groups by representing them as homotopy equivalences of graphs. The space of all such homotopy equivalences, called Outer space, is a contractible space on which the group of outer automorphisms of a free group acts properlydiscontinuously. Analyzing the quotient of Outer space by this action has yielded a significant amount of algebraic information about this group over the years. More recently,work of Maxim Kontsevich has related the rational homology of this quotient to the homology of a certain infinite-dimensional Lie algebra, consisting of derivations of the free Lie algebra. Kontsevich also defined variations of this infinite-dimensional Lie algebra whose homologies are related to the homology of mapping class groups and toKontsevich's graph homology, which is a homology theory containing various 3-manifold and knot invariants. The Principal Investigator will use this point of view to obtain new information about invariants of these groups of classical interest to topologists; conversely, topological methods will be applied to obtain new information about these Lie algebras and their homology. In particular, new Lie bi-algebra structures on the chain complexes used tocompute graph homology, recently discovered by Jim Conant the the Principal Investivator, will be investivated. Finally, the local geometry of Outer space will be studied. Certain neighborhoods in Outer space are non-positively curved metric spaces, and can be identified with spaces of finite labelled metric trees. Such trees are of interest in many branches of science, including biology, where they appear as evolutionary trees produced from DNA data. Another aim of this project is to determine efficient algorithms for finding geodesics in this space, and to continue from there to develop a program of applying statistical methods to this space of trees. Graphs and trees appear in many contexts in the sciences and in mathematics. In Geometric Group Theory a space of graphs known as Outer space, originally defined by Marc Culler and the Principal Investigator, has been used extensively to study the group of outer automorphisms of a finitely generated free group. In this project the Principal Investigator will continue to study how the geometry of this space is related to algebraic properties of the group of automorphisms. A new perspective on this problem is provided by work of M. Kontsevich; motivated by considerations from physics and symplectic geometry, Kontsevich defined various abstract infinite-dimensional algebraic structures whose invariants are closely related to invariants of Outer space. Finally, small regions of Outer space can be identified with spaces of finite trees. Such trees are used as a tool by evolutionary biologists, and another focus of this project is understanding the local geometry of these regions in a very concrete, algorithmic way which can be applied to problems inmolecular phylogenetics.