The Topology of Manifolds of Dimensions 3 and 4

University Of Texas At Austin, Austin TX

Investigators

Abstract

Abstract Award: DMS 0229035 Principal Investigator: Cameron Gordon The project is a conference in 3- and 4-dimensional topology. Satisfactory accounts of manifolds of dimension greater than or equal to 5 were achieved by the late 1960's. Since then, attention has naturally focused on dimensions 3 and 4. Despite dramatic advances (Thurston, Freedman, Donaldson, Jones, Witten...), many major problems in these low dimensions remain unsolved, for example the classical 3-dimensional Poincare Conjecture and its smooth 4-dimensional analog. The topological classification of 1-connected 4-manifolds was achieved by Freedman in the early 1980's, but in the smooth case there is not even a conjectural picture. By contrast, Thurston's Geometrization Conjecture provides a beautiful and coherent description of all 3-manifolds, but is far from established. In the mid 1980's, Jones' discovery of his polynomial link invariant, together with work of Witten, led to the introduction into 3-dimensional topology of methods from quantum physics. The current situation in low-dimensional topology is that there are many different directions and methods, but little understanding of the relations between them. Thus in dimension 3 we have hyperbolic geometry, foliations and laminations, normal surfaces, combinatorial geometric methods, quantum invariants, finite type invariants, Floer homology,..., while in dimension 4 there are the Donaldson and Seiberg-Witten theories, symplectic structures,.... The aim of the conference is to address these topics, and the connections between them, and to provide a forum for interaction and exchange of ideas between experts in the different areas. Geometric topology aims to understand n-dimensional manifolds, which are objects that locally look like ordinary n-dimensional Euclidean space (whose points are described by n co-ordinates), but whose global structure might be quite complicated. The cases of dimensions 3 and 4 are particularly interesting, being the dimensions of our spatial and spatial-temporal universes, and it is a striking fact that it is precisely these dimensions that are mathematically anomalous. There have been many developments and considerable progress in the fields of 3- and 4-dimensional topology over the last twenty-five years, but, in both dimensions, a complete picture is still lacking. One of the aims of the conference is to encourage interaction and collaboration between experts in the several different aspects and techniques that are currently being pursued in low-dimensional topology, and in particular between those people working in dimension 3 and those in dimension 4, where the methods in the two areas tend to be quite different, but where there are several hints of connections between them. The current disparate state of the subject makes it difficult for people beginning research in low-dimensional topology to get a good overview of the area, and so by encouraging the participation of graduate students and postdoctoral researchers, we intend that the conference should also provide an opportunity for young researchers to get a broad perspective of the present state of knowledge, directions begin pursued, and the main open problems. The invited speakers include some of the world's leading 3- and 4-dimensional topologists, whose expertise together covers all the major aspects of the two fields.