Geometry and Topology of Knots and Manifolds

Brandeis University, Waltham MA

Investigators

Abstract

DMS-0204386 Daniel 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.