Finanzierung: Europäische Union
Programm: Research Training Networks
Laufzeit:1.6.2000 - 31.5.2004
Zusammenfassung: EDGE aims to encourage and facilitate research and training in major areas of differential geometry, which is a vibrant and central topic in pure mathematics today. A significant theme which unites the areas that are the subject of this endeavour is the interface with other disciplines, both pure (topology, algebraic geometry) and applied (mathematical physics, especially gauge theory and string theory). The members of EDGE are geometers in mathematical centres spreading among most European countries. These centres are grouped into nine geographical nodes which are responsible for the management of joint research projects and for the training of young researchers through exchange between the EDGE groups.
The following are some of the common mathematical themes that underlie and unify the tasks to be addressed by EDGE.
The study of Riemannian geometry in the complex setting often yields strong and interesting results that can have an impact both on Riemannian geometry and algebraic geometry. For example Kähler-Einstein metrics and minimal submanifolds in Kähler manifolds are two subjects where the interplay between real methods from PDE and complex geometry yields deep insights.
Another unifying theme is the use of analytical and differential-geometric methods in attacking problems whose origin is not in differential geometry per se. These methods will be used by researchers throughout the network to investigate a wide variety of problems in related areas of mathematics including topology, algebraic geometry, and mathematical physics. In algebraic geometry, for example, there are a number of problems that are best attacked with `transcendental methods'. In some cases, the research concerns correspondences between differential-geometric and algebraic-geometric objects (as in the Hitchin-Kobayashi correspondence and its generalizations).
Symplectic geometry is a part of geometry where `almost-complex' methods already play a large role, and this area forms an integral part of the proposed research. Moreover `moment-map' ideas play a very significant role in other parts of the research programme, for example in the construction of extremal Kähler metrics and quaternionic-Kähler and hyper-Kähler metrics. In addition, infinite-dimensional analogues of the correspondence between symplectic and stable quotients provide a good conceptual framework for the understanding of many phenomena in gauge theory and complex differential geometry. The aim of exploiting this conceptual framework to the full thus unifies several of the research tasks.
Twistor methods give a correspondence between holomorphic geometry and low-dimensional conformal geometry, and also allow the use of complex methods in the study of quaternionic geometry, (parts of) gauge theory and integrable systems, all of which are subjects included in this research proposal.
The research to be undertaken by the proposed network is in pure mathematics, but much of it is closely related to, and inspired by, developments in supersymmetric quantum field theory and superstring theory. The input from these areas takes the form of conjectures (especially in Donaldson-Yang-Mills theory and in Seiberg-Witten theory) as well as new geometric structures (Frobenius manifolds, special Kähler geometry) which often turn out to be of interest in complex differential geometry. Conversely, mathematical developments in these subjects have feedback in physics. This is well-known for gauge theory, but it also applies to quaternionic geometry and exotic holonomy, which are of increasing interest in string theory via D-branes.
Mitarbeiter: Dr. G. Bande, Dr. P. Ghiggini, Dr. J. Kedra
Ehemalige Mitarbeiter: Dr. J. Amoros, Dr. P.-A. Nagy, S. Chaichenets