Geometry and Topology of Knots and Manifolds

DMS-0204386Daniel Ruberman，Jerome Levine和Kiyoshi Igusa Brandeis大学的拓扑组建议研究纽结理论和3-流形，规范理论和4-流形，以及流形族的高阶Reidemister挠率。Jerome Levine计划继续他对三维流形的有限类型不变量的研究。特别是，他计划研究对最近一篇论文中提出的一种新的外科理论的图形解释。他还将研究与边界链不变量有关的各种问题，如签名型不变量以及Farber不变量与非对易Reidemister挠率之间的关系。丹尼尔·鲁伯曼的研究使用规范理论来研究4维流形的几何和拓扑。他将扩展他的程序，利用Seiberg-Witten理论和Yang-Mills理论来探索4-流形的微分同胚族和4-流形上的正标量曲率度量。进一步的工作探讨了黎曼几何的有限性定理在4维空间中是否成立。Seiberg-Witten和Donaldson理论也将被用来研究一类简单但重要的非单连通4-流形。井上清的工作将他对高阶Franz-Reidemister挠率的研究扩展到了复杂情形和各种新的图空间。他将用高阶Franz-Reidemister(FR)挠率来解释Miller-Morita-Mumford类。我们的研究包括几何和拓扑学中的一些主题。其中一个是拓扑学和代数之间的相互作用，另一个是拓扑学和理论物理之间的相互作用。为了研究空间或其他拓扑对象，如纽结，我们将空间的某些方面编码为熟悉的代数对象，如矩阵或多项式。一些用于研究三维物体的代数与物理学家提出的理解量子场论的想法有关，特别是费曼积分。研究这种相互作用有助于我们理解我们所生活的三维世界。我们的部分工作是关于空间的演化--研究空间随着某些参数的变化而变化的方式。这方面的一些研究开发了一些代数工具，这些工具适用于有许多独立参数的情况。另一个组件研究4维空间和几何的演化，使用规范理论中出现的技术，规范理论是理论物理中的基本几何工具。

DMS-0204386Daniel Ruberman, Jerome Levine, and Kiyoshi Igusa The topology group at Brandeis University proposes research on knot theory and 3-manifolds, gauge-theory and 4-manifolds, and on higher Reidemeister torsion of families of manifolds. Jerome Levine plans to continue his study of finite type invariants of 3-manifolds. In particular, he plans to investigate a graphical interpretation of a new surgery-theoretic theory proposed in a recent paper. He also will investigate various questions concerning boundary link invariants such as the signature-type invariant and the relationship between the Farber invariant and non-commutative Reidemeister torsion. Daniel Ruberman's research uses gauge theory to investigate the geometry and topology of 4-dimensional manifolds. He will extend his program of using Seiberg-Witten theory and Yang-Mills theory to explore families of diffeomorphisms of 4-manifolds and positive scalar curvature metrics on a 4-manifold. Further work explores whether finiteness theorems of Riemannian geometry hold in dimension 4. Seiberg-Witten and Donaldson theory will also be used to investigate a simple but important class of non-simply-connected 4-manifolds. Kiyoshi Igusa's work extends his study of higher Franz-Reidemeister torsion to the complex case and to various new spaces of graphs. He will use this to interpret the Miller-Morita-Mumford classes in terms of the higher Franz-Reidemeister (FR) torsion.Our research encompasses a number of themes in geometry and topology. One of these is the interplay between topology and algebra, and another is the interplay between topology and theoretical physics. To study spaces or other topological objects such as knots, we encode some aspects of the space into familiar algebraic objects such as matrices or polynomials. Some of the algebra used to study 3-dimensional objects is connected with ideas developed by physicists to understand quantum field theory, especially the Feynmann integral. Investigating this interplay contributes to our understanding of the 3-dimensional world in which we live. Part of our work is concerned with the evolution of spaces--the study of the way in which they change as some parameter is varied. Some of the research in this direction develops algebraic tools which apply in the case when there are many independent parameters. Another component studies the evolution of 4-dimensional spaces and geometries, using techniques arising in gauge theory, a fundamental geometrical tool in theoretical physics.