3-Manifolds After Perelman; March 2006; Edinburgh, UK

University Of Texas At Austin, Austin TX

Investigators

Abstract

Abstract Award: DMS-0601251 Principal Investigator: Cameron McA. Gordon, Alan W. Reid The project is a workshop on 3-dimensional topology that will be held at the International Centre for Mathematical Sciences in Edinburgh, UK, in March, 2006. Thurston's Geometrization Conjecture, which has guided research in 3-dimensional topology for almost three decades, has now been established, and consequently there is a need to set out a program for future research in the subject. This is the objective of the workshop. The program will be centered around five series of three 1-hour lectures, given by Jeff Brock, Dave Gabai, Marc Lackenby, Peter Ozsvath, and Peter Shalen. There will also be nine additional invited 1-hour talks by other leading researchers. Among the recent important developments in 3-dimensional topology that will be discussed, with a view to illuminating the theory of 3-manifolds beyond Geometrization, are: the proofs of the Tameness and Ending Lamination Conjectures, Heegaard Floer homology theory, various approaches to the virtual Haken Conjecture, and the connection between 3-dimensional topology and the theory of word hyperbolic groups. 3-dimensional topology is a large and active subject within the general field of pure mathematics. Its goal is the study of all possible 3-dimensional "spaces", and of the various objects that can be situated in them, such as knots. Over the last 30 years or so this subject has seen many exciting developments, revealing connections to many other branches of mathematics as well as quantum physics and (in the case of knots) even to the behavior of DNA. The guiding light of the subject during this period has been the Geometrization Conjecture, put forward William Thurston in the mid 1970's, which asserts that hyperbolic geometry, the non-euclidean geometry discovered by Lobachevsky in the 19th century, plays a key role in the description of these 3-dimensional spaces. This conjecture was recently proved by Grisha Perelman. It seems timely, then, to take stock of the subject, identifying the major problems that remain and indicating potential techniques and directions for future work. The core of the program will be five series of three 1-hour lectures, given by five internationally renowned experts in the subject. It is expected that this will make the workshop especially useful to young researchers. The conference web site is http://www.icms.org.uk/meetings/2006/3-manifolds/index.html.