University of Durham --- Department of Mathematical Sciences

Yorkshire and Durham Geometry Days

A Yorkshire and Durham Geometry Day will take place on Friday February 24, 2017 in the Department of Mathematical Sciences at Durham University. The timetable is as follows:

11:00 Coffee Mathematics Department Common Room CM211

11:45 David Cushing (Durham) in CM301

"Bakry Emery Curvature Functions of Graphs"
Abstract: We study the Bakry-Emery curvature function \(K_{G,x}: [0,\infty )\rightarrow \mathbb{R}\) of a vertex \(x\) in a locally finite graph G systematically. Here \(K_{G,x}(N)\) is defined as the optimal curvature lower bound \(K\) in the Bakry-\'Emery curvature-dimension inequality \(CD(K,N)\) that \(x\) satisfies. We prove the curvature functions of the Cartesian product of two graphs \(G_{1},G_{2}\) equal an abstract product of curvature functions of \(G_{1},G_{2}\). We relate the curvature functions of \(G\) with various spectral properties of (weighted) graphs constructed from local structures of \(G\). We explore the curvature functions of Cayley graphs, strongly regular graphs, and many particular (families of) examples including Johnson graphs and complete bipartite graphs. We will also demonstrate a piece of software developed in order to calculate the curvature of graphs.

12:45 Lunch break

13:45 Ben Lambert (UCL) in CM301

"Maximal Hypersurfaces with boundary conditions"

Abstract: We construct maximal surfaces with Neumann boundary conditions in Minkowski space using mean curvature flow. In particular we find curvature conditions on a boundary manifold so that mean curvature flow may be shown to exist for all time, and give conditions under which the maximal hypersurfaces are stable under the flow.

15:00 Andrea Mondino (Warwick) in CM301

"Isoperimetric inequalities in non-smooth spaces with Ricci curvature bounded below "

Abstract: In the first part of the talk I will recall the notion of Ricci curvature lower bounds in non-smooth spaces via optimal transport introduced by Lott-Sturm-Villani, then I will discuss some basic properties of such spaces and finally I will discuss the non-smooth extension of the celebrated Levy-Gromov isoperimetric inequality.

16:00 Tea break in CM211

17:00 Luc Nguyen (Oxford) in CM301

"Recent progress on nonlinear Yamabe problems in conformal geometry"
Abstract: The Yamabe problem asks to find in a given conformal class of metrics a metric of constant scalar curvature. I will discuss nonlinear analogues where the scalar curvature, which is the trace of the Ricci curvature tensor, is replaced by other functions of the eigenvalues of the Ricci curvature tensor.

18:15 Leave for dinner at the Lebaneat on 69 Claypath 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

Derek Harland & Gerasim Kokarev, University of Leeds

Ian McIntosh & Chris Wood, University of York