University of Durham --- Department of Mathematics

Yorkshire and Durham Geometry Days

A Yorkshire and Durham Geometry Day will take place on Friday March 2, 2018 in the Department of Mathematics at Durham University. The timetable is as follows:

10:30 Coffee in the Mathematics Department Common Room CM211

11:00 Nicolas Juillet (Universite de Strasbourg) in CM301

"Deformation of singular spaces"
Abstract: Gigli and Mantegazza have observed how optimal transport and heat diffusion allow to describe the direction of the Ricci flow uniquely from the metric aspects of Riemannian manifolds. Their goal is to reformulate the Ricci flow so that this flow also makes sense for metric spaces. I will present investigations and results obtained with Matthias Erbar (University of Bonn) that concerns some non-Riemannian limits of Riemannian manifolds.

12:00 Lunch break

2:00 Alessio Figalli (ETH Zuerich) in CM301

"The De Giorgi conjecture for the boundary reaction terms in dimension 4+1"

Abstract: The famous De Giorgi conjecture for the Allen-Cahn equation states that global monotone solutions are 1D, at least if the dimension is less than 9. This conjecture is motivated by classical results about the structure of global minimal surfaces. The analogue of this conjecture in the half-space with the boundary reaction terms can be reduced to study a fractional Allen-Cahn equation in the whole space. In this talk I will present a recent result with Joaquim Serra, where we prove the validity of the De Giorgi conjecture for the boundary reaction terms in dimension 4+1.

3:10 Lucia Scardia (University of Bath) in CM301

"Equilibrium measures for nonlocal energies: The effect of anisotropy"

Nonlocal energies are continuum models for large systems of particles with long-range interactions. Under the assumption that the interaction potential is radially symmetric, several authors have investigated qualitative properties of energy minimisers‚Äč. But what can be said in the case of anisotropic kernels? Motivated by the example of dislocation interactions in materials science, we pushed the methods developed for nonlocal energies beyond the case of radially symmetric potentials, and discovered surprising connections with random matrices and fluid dynamics.

4:10 Tea break in CM211

5:00 Josh Cork (University of Leeds) in CM301

"Calorons, the rotation map, and cyclic symmetries"

Abstract: There is a lot of interest in understanding the moduli spaces of solutions to variational problems, and one useful approach is to consider fixed points under isometries. This has been utilised in many cases relatively successfully for isometries arising from the underlying manifold. Calorons are finite-action, anti-self-dual connections on S^1\times R^3, and the finite action condition necessarily leads to a set of paired topological boundary data, called masses and charges, which in turn determine the value of the action. For each set of boundary data, there exists an isometry of moduli spaces called the rotation map, which plays the role of interchanging the masses and charges. In this talk we shall give a reasonable introduction to calorons and the isometries on the moduli space, including the rotation map. We will then consider fixed points of various "rotation map cyclic groups", in the particular case where the pairs of masses and charges are equal, by utilising the monad construction, and Nahm transform for calorons.

5:30 Daniel Ballesteros-Chavez (Durham University) in CM301

"Prescribed k-curvature of convex closed hypersurfaces in H^n and S^n : a fully nonlinear elliptic problem"

Abstract: We will present a priori C^2 bounds for convex hypersurfaces with prescribed k-symmetric curvatures in two ambient model spaces of constant sectional curvature. These serve to prove existence thereof in the case of a barrier assumption on the prescription of k -curvature. Given the C^2 - estimate, this follows from Evans - Krylov C^{2+a} regularity and an iteration argument, where the barrier gives C^0 and C^1 estimates.

6:15 Leave from CM211 for dinner at Lebaneat at 69 Claypath

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 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

Derek Harland & Gerasim Kokarev, University of Leeds

Ian McIntosh & Chris Wood, University of York