The current members of the geometry group at York are:
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Dr Chris Wood Dr Ian McIntosh Dr Brent Everitt |
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The interests of the group members are as follows. For more details and lists of publications, please follow the links to individual homepages above.
- Dr Chris Wood
Chris's interest lies in the study of harmonic mappings between Riemannian manifolds. In particular, Chris studies Riemannian G-structures from a variational viewpoint, by looking at the behaviour of the energy functional on the space of cross-sections of an associated homogeneous fibre. This involves generalizing the notion of a harmonic map to a harmonic section. Applications to submanifold theory, distributions on manifolds, and almost-complex structures are currently being researched, with particular interest in questions of stability.
- Dr Ian McIntosh
Ian works on the application of integrable systems to surface theory. Integrable systems can be found in a surprising variety of different types of surfaces, for example:
- minimal and harmonic surfaces in space forms,
- CMC surfaces in Euclidean, hyperbolic and spherical 3-space,
- Willmore surfaces,
- conformally immersed surfaces in the 4-sphere,
- Lagrangian surfaces in Hermitian symmetric spaces.
Right now Ian is interested in the role integrable systems play in the construction of conformally immersed tori in the 4-sphere, and the prospects for generalizing this to n-spheres. The mathematics involved is an interesting mixture of infinite dimensional Lie group theory, classical geometry of surfaces and algebraic curve theory.
- Dr Brent Everitt
Brent is interested in (differentiable) manifolds equipped with so-called geometric structures, which loosely means that one can do geometry locally on the manifold. What is meant by geometry can be interpreted broadly (for example the geometry of the n-sphere, the geometry of the n-dimensional Heisenberg group, ...), but it is the hyperbolic geometries that provide the richest theory. The hyperbolic geometries are nothing other than the rank one symmetric spaces, ie: the n-dimensional real hyperbolic space, the n-dimensional complex hyperbolic space, the n-dimensional quaternionic hyperbolic space, and the octonionic hyperbolic plane.
Brent studies such things through group theoretic spectacles, using the observation that any such manifold has universal cover the appropriate geometry, with fundamental group acting freely and properly by isometries on this universal cover. Certain powerful rigidity results, due to Margulis, Mostow, Thurston, Gromov and others, allow us to identify the manifold with its fundamental group (except in the 2-dimensional real case). Thus to study the manifolds, one studies the groups. It is this two way street between the theory of discrete groups and the theory of geometric manifolds that Brent travels along.
Yorkshire Durham Geometry Days
We are part of the three way Durham-Leeds-York network of LMS sponsored seminars called Yorkshire Durham Geometry Days. Click the link to see the programme of past and future events.
| Related sites for geometry: |
email: im7 (append @york.ac.uk from outside the University of York).