UCL Geometry and Topology Days
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UCL Geometry and Topology Days

University College London, 20th February, 20th March, 31st May 2013

To celebrate the appointments of new geometers at UCL (Jason Lotay, Michael Singer, Felix Schulze, Chris Wendl, Henry Wilton and Jonny Evans), the London Mathematical Society have funded three half-day sessions of geometry-themed talks.

The third (31st May 2013) will have three talks focused on Geometric Analysis, with speakers: Felix Schulze (UCL), Tobias Lamm (Karlsruhe Institute for Technology) and Peter Topping (University of Warwick). The event will be followed by a drinks reception and dinner.

The first (Riemannian Geometry, 20th February) and second (Symplectic/Contact Geometry, 20th March) were extremely successful, with guest speakers Olivier Biquard (ENS Paris) and Hansjörg Geiges (Universität zu Köln).

If you are interested in attending this event or would like further information, please send an email to Dr Felix Schulze (f.schulze@ucl.ac.uk).

Third Event, Geometric Analysis (Friday 31st May 2013)

The talks will take place in Room 500 in the Maths Department at UCL (25 Gordon St).

Abstracts:

Tobias Lamm (Karlsruhe Institute of Technology) - Willmore surfaces and conformal immersions: In this talk I try to explain two existence results for Willmore surfaces with prescribed area in a Riemannian manifold. They rely on an interesting generalization of a classical result of Codazzi which is due to DeLellis and Müller. Moreover, I will introduce W2,2 conformal immersions and I will explain how they can be used to obtain extensions of the result of DeLellis and Müller. These results rely on joint works with Jan Metzger, Huy T. Nguyen and Felix Schulze.

Felix Schulze (UCL) - The half-space property and entire positive minimal graphs in M x R: In this talk I will discuss the following two results. First, I will show that a properly immersed minimal hypersurface in M × R+ equals some M × {c} when M is a complete, recurrent n-dimensional Riemannian manifold with bounded curvature. Second, if M is not necessarily recurrent but has non-negative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over M. This is joint work with Harold Rosenberg and Joel Spruck.

Peter Topping (University of Warwick) - Uniqueness of instantaneously complete Ricci flows: I will explain the very recent completion of the well-posedness theory for Ricci flows on surfaces starting with a completely general metric, not necessarily complete or of bounded curvature. There is always exactly one flow that is complete for positive times. I will also explain some unusual features of these flows.

Previous events

20 March: Symplectic Topology

20 February: Riemannian Geometry

The Hopf fibration.

Image courtesy of: Imaging Technology Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Illinois, USA.

The picture shows some fibres of the Hopf fibration, a fibring of the three-sphere by linked circles. The space of fibres is the two-sphere and the corresponding map from the three-sphere to the two-sphere is the simplest map with nonzero Hopf invariant. See this page for a beautiful explanation and accompanying video.

The fibres chosen here all live on the (stereographically projected) Clifford torus. The Lawson conjecture states that this is the only minimally embedded torus in the three-sphere (up to isometry) - this conjecture was recently proved by S. Brendle, after being open for more than forty years.

It also arises as a global minimiser for the Willmore energy functional on embedded surfaces in the three-sphere of genus at least one. This Willmore conjecture had been open since 1965 and was recently proved by Marques and Neves.