Graphs, Trees and Geometric Group Theory

karen vogtmann这个项目继续发展一套思想，这些思想开始于试图通过将自由群表示为图的同伦等价来理解自同构群。所有这些同伦等价的空间称为外空间，是自由群的外自同构群在其上不连续地适当作用的可收缩空间。多年来，通过这一行动分析外层空间的商已经产生了大量关于这一群体的代数信息。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，这个无穷维李代数是由自由李代数的导数组成的。Kontsevich还定义了这个无限维李代数的变体，这些变体的同调与映射类群的同调和Kontsevich的图同调有关，后者是一个包含各种3流形和结不变量的同调理论。首席研究员将利用这一观点获得拓扑学家感兴趣的经典群的不变量的新信息；相反，拓扑方法将被应用于获得关于这些李代数及其同调的新信息。特别地，将研究由首席研究员Jim Conant最近发现的用于计算图同调的链配合物上的新的李双代数结构。最后，研究了外空间的局部几何。外空间中的某些邻域是非正弯曲度量空间，可以用有限标记度量树的空间来识别。这样的树在包括生物学在内的许多科学分支中都引起了人们的兴趣，在生物学中，它们是由DNA数据产生的进化树。这个项目的另一个目的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于这个树木空间的程序。图和树出现在科学和数学的许多语境中。在几何群论中，被称为外空间的图空间，最初由Marc Culler和首席研究员定义，已被广泛用于研究有限生成自由群的外自同构群。在这个项目中，首席研究员将继续研究这个空间的几何如何与自同构群的代数性质相关。康采维奇的工作为这个问题提供了一个新的视角；出于对物理和辛几何的考虑，Kontsevich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的空间来识别。这样的树被进化生物学家用作一种工具，这个项目的另一个重点是用一种非常具体的算法方法来理解这些区域的局部几何形状，这种方法可以应用于分子系统发育问题。

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.