University of Durham --- Department of Mathematical Sciences

Yorkshire and Durham Geometry Days

A Yorkshire and Durham Geometry Day will take place on Friday February 5, 2016 in the Department of Mathematical Sciences at Durham University. The timetable is as follows:

10:30 Coffee Mathematics Department Common Room CM211

11:00 Katie Gittins (Bristol) in CG83

"Shape optimisation of the Dirichlet and Neumann eigenvalues of the Laplacian"
Abstract: We consider shape optimisation problems for the eigenvalues of the Laplacian on open sets of $\R^m, m \geq 2$ of finite measure. For $k \in \N$, the aim is to optimise the $k$'th eigenvalue of the Laplacian over a given class of domains subject to certain boundary conditions. Despite being an incredibly active area of research throughout the last century, there are still a wealth of interesting open problems, even for small $k$. We focus our attention on the minimisation of the Dirichlet eigenvalues of the Laplacian among sets of prescribed measure. We review some known results for this problem and then compare with the analogous problem of maximising the Neumann eigenvalues of the Laplacian among sets of prescribed measure.

12:00 Lunch break

1:00 Melanie Rupflin (Oxford) in CG83

"Flowing to minimal surfaces"

For maps from surfaces there is a close connection between the area functional and Dirichlet energy and thus also between their critical points. As such, one way to try to find minimal surfaces is to consider a gradient flow of the Dirichlet energy, which not only evolves a map but also the domain metric in order to find a map that is not only harmonic but also (weakly) conformal and thus a (branched) minimal immersion. In this talk I will discuss the construction of such a flow, the Teichmueller harmonic map flow, and explain in particular how this flow decomposes any given initial map into one or more branched minimal immersions.

2:00 Otis Chodosh (Cambridge) in CG83

"Area-minimizing surfaces in asymptotically flat three-manifolds"

Abstract: I will discuss recent work with Michael Eichmair . We show that an asymptotically flat three-manifold with non-negative scalar curvature cannot admit an unbounded area minimizing surface unless the ambient space is flat. This has consequences for the behavior of isoperimetric regions in such manifolds and the classification of asymptotically flat static three manifolds.

3:00 Tea break in CM211

4:00 Stefan Wenger (Fribourg) in CG83

"Minimal discs in metric spaces and applications to large scale geometry"
Abstract: The classical Problem of Plateau asks to find a disc-type surface of minimal area with prescribed Jordan boundary. This problem was first solved by Douglas and Rado in the setting of Euclidean space and then extended by Morrey to the setting of Riemannian manifolds. In this talk, I will present a generalization to metric spaces. Precisely, I will show that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization (in a weak sense). If the space admits a local quadratic isoperimetric inequality for curves then such a disc is locally Hölder continuous in the interior and continuous up to the boundary. I will then discuss some applications to the large scale geometry of groups and spaces with a quadratic isoperimetric inequality. The talk is based on joint work with Alexander Lytchak.

5:30 Leave for dinner at the Zizi from CM211

Travel: Durham is easy to get to by car and train, and so is the Department of Mathematics, located on the Science Site. Click here for relevant information.

Yorkshire and Durham Geometry Days are jointly organised by the Universities of Durham, Leeds and York, and occur at a frequency of three meetings per academic year. Financial support is provided by the London Mathematical Society through a Scheme 3 grant, currently administered by the University of York. Additional support is provided by the Department of Mathematics, Durham University.

The local organizers are:

John Bolton & Wilhelm Klingenberg, University of Durham

Martin Speight & John Wood, University of Leeds

Ian McIntosh & Chris Wood, University of York