# 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 HÃ¶lder 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