The sixth William Rowan Hamilton Geometry and Topology Workshop is a three-day, directed workshop on Knots, Surfaces and Three-Manifolds, to be held at the Hamilton Mathematical Institute (HMI) in Dublin, Ireland September 2-4, 2010. The purpose of the William Rowan Hamilton Geometry and Topology Workshop is to investigate common themes and techniques among significant areas of current research in geometry and topology and to support junior researchers interested in these areas. This years workshop will bring together leading researchers in geometry and topology who have a special interest in problems related to Knots, Surfaces and Three-Manifolds and will investigate a number of important questions at the forefront of research in these areas. The confluence of expertise from different areas will result in new collaborative projects, and in broadening the research horizons of the participants.
Among the main topics of the workshop will be the virtual Haken conjecture, that every closed hyperbolic 3-manifold has a finite cover that contains a closed incompressible surface. The conjecture has motivated extensive work of many researchers in recent times, and is one of the main problems in the area. A surge of new and exciting ideas, in particular Kahn and Markovic's proof of the surface subgroup conjecture, Wise's work on quasiconvex heirarchies, and Agol's virtual fibering criterion, has provided new avenues to tackle this problem and there is reason to believe that a solution may well be in sight. The workshop will also concentrate on recent advances in low-dimensional topology made by researchers in Heegaard Floer theory. We will discuss open questions in the field including the Berge conjecture which gives a conjectural list of those knots in the three-sphere which admit a Lens-space surgery. Another topic will be the recently announced proof by Guilfoyle and Klingenberg of the Caratheodory conjecture, that any closed convex surface in 3-dimensional Euclidean space must have at least 2 umbilic points. Through sharing new techniques and insights, it is hoped that progress can be made on a number of these important questions.
held at the Hamilton Mathematical Institute (HMI) in Dublin, Ireland September 2-4, 2010. This was a three-day, directed workshop on Knots, Surfaces and Three-Manifolds. Activities and their Intended Impacts. The workshop brought together some of the leading researchers in geometry and topology who have a special interest in problems related to Knots, Surfaces and Three-Manifolds. This is an important topic of current research interest in the field of geometry and topology and the confluence of expertise from different areas supported new collaborative projects, and in broadened the research horizons of the participants. The format consisted of background talks in the morning, followed by a question-and-answer session, and then afternoon lectures which will describe current research. On the last day, there will be a problem session in which participants were encouraged to describe parallels that they observed between the problems and techniques used in the various fields. Intellectual Merit. The workshop investigated common themes and techniques among different areas of research in knots, surfaces and three-manifolds. This is an important field of study which considers a general theory of geometry of both abstract mathematical spaces and the physical spaces such as our three-dimensional world and four dimensional space-time. The confluence of expertise from different areas at the workshop supported new collaborative projects, and in broadened the research horizons of the participants. Broader Impacts. Immediate broader impacts include the generation of a more well-defined set of research directions and open problems in a number of related fields for the next generation of young researchers to work on. The workshop also supports strong links between the US and rest of the international research community and is a major annual forum for communication between the U.S. and European research communities.
