Graphs, Trees and Geometric Group Theory

DMS-0204185 Karen Vogtman这个项目继续发展一套想法，开始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。所有这样的同伦等价的空间，称为外层空间，是一个可收缩的空间，在这个空间上，自由群的外层自同构群适当地不连续地作用。 多年来，通过这种作用分析外层空间的商已经产生了大量关于这个群体的代数信息。 最近，马克西姆·孔采维奇（Maxim Kontsevich）的工作将这个商的有理同调与由自由李代数的导子组成的某个无限维李代数的同调联系起来。 Kontsevich还定义了这个无限维李代数的变体，其同调与映射类群的同调和Kontsevich的图同调有关，这是一个包含各种3-流形和结不变量的同调理论。 主要研究者将使用这一观点，以获得新的信息不变量的这些群体的经典利益的拓扑;相反，拓扑方法将被应用于获得新的信息，这些李代数及其同源性。 特别是，新的李双代数结构上的链复合体用于计算图的同源性，最近发现的吉姆科南特的主要Investivator，将investivative。最后，将研究外层空间的局部几何。 外层空间中的某些邻域是非正曲度量空间，并且可以与有限标号度量树空间相一致。 这种树在许多科学分支中都很有意义，包括生物学，在生物学中，它们是从DNA数据中产生的进化树。该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，并继续开发将统计方法应用于这个树空间的程序。 图和树出现在科学和数学的许多背景中。 在几何群论中，最初由Marc Culler和首席研究员定义的图形空间称为外层空间，已被广泛用于研究一个由图生成的自由群的外自同构群。 在这个项目中，主要研究者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。M.孔采维奇;出于物理学和辛几何的考虑，Kontsevich定义了各种抽象的无限维代数结构，其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的空间来识别。 这种树被进化生物学家用作工具，该项目的另一个重点是以一种非常具体的算法方式理解这些区域的局部几何形状，这种方法可以应用于分子遗传学中的问题。

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.