EUROPEAN DIFFERENTIAL GEOMETRY ENDEAVOUR
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.
Comments and suggestions are very welcome.
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