Tue, 14 Oct 2014

14:30 - 15:30
L6

The structure of graphs which are locally indistinguishable from a lattice.

David Ellis
(Queen Mary University of London)
Abstract

We study the properties of finite graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to some fixed graph $F$. (Such a graph is said to be $r$-locally-$F$.) This is a natural extension of the study of regular graphs, and of the study of graphs of constant link. We focus on the case where $F$ is $\mathbb{L}^d$, the $d$-dimensional integer lattice. We obtain a characterisation of all the finite graphs in which the ball of radius $3$ around each vertex is isomorphic to the ball of radius $3$ in $\mathbb{L}^d$, for each integer $d$. These graphs have a very rigidly proscribed global structure, much more so than that of $(2d)$-regular graphs. (They can be viewed as quotient lattices in certain 'flat orbifolds'.) Our results are best possible in the sense that '3' cannot be replaced with '2'. Our proofs use a mixture of techniques and results from combinatorics, algebraic topology and group theory. We will also discuss some results and open problems on the properties of a random n-vertex graph which is $r$-locally-$F$. This is all joint work with Itai Benjamini (Weizmann Institute of Science). 

Diffusion of the dead - mathematics, zombies and the spread of infection

No doubt you have found yourself screaming at the stupidity of characters in a horror movie. You would never have walked into that room alone, you would always make sure your weapon was loaded and you would certainly always ensure a monster was dead before returning to the corpse!

But what are the real rules that you should live by if the dead should begin to rise? Dr Thomas Woolley and coauthors from the Mathematical Institute, University of Oxford believe they have come up with the answer:

Thu, 04 Dec 2014

12:00 - 13:00
L4

Higher regularity of the free boundary in the elliptic thin obstacle problem

Wenhui Shi
(Bonn University)
Abstract

In this talk, I will describe how to use the partial hodograph-Legendre transformation to show the analyticity of the free boundary in the elliptic thin obstacle problem. In particular, I will discuss the invertibility of this transformation and show that the resulting fully nonlinear PDE has a subelliptic structure. This is based on a joint work with Herbert Koch and Arshak Petrosyan.

Tue, 21 Oct 2014

12:45 - 13:45
C4

TBA

Alexander Vervuurt, Jochen Kursawe, Linus Schumacher
(Mathematical Institute, Oxford)
Thu, 27 Nov 2014

12:00 - 13:00
L4

Interface motion in ill-posed diffusion equations

Michael Helmers
(Bonn University)
Abstract
We consider a discrete nonlinear diffusion equation with bistable nonlinearity. The formal continuum limit of this problem is an
ill-posed PDE, thus any limit dynamics might feature measure-valued solutions, phases interfaces, and hysteretic interface motion.
Based on numerical simulations, we first discuss the phenomena that occur for different types of initial. Then we focus on the case of
interfaces with non-trivial dynamics and study the rigorous passage to the limit for a piecewise affine nonlinearity.
Thu, 06 Nov 2014

12:00 - 13:00
L4

Towards an effective theory for nematic elastomers in a membrane limit

Paul Plucinsky
(Caltech)
Abstract
 

For nematic elastomers in a membrane limit, one expects in the elastic theory an interplay of material and structural non-linearities. For instance, nematic elastomer material has an associated anisotropy which allows for the formation of microstructure via nematic reorientation under deformation. Furthermore, polymeric membrane type structures (of which nematic elastomer membranes are a type) often wrinkle under applied deformations or tractions to avoid compressive stresses. An interesting question which motivates this study is whether the formation of microstructure can suppress wrinkling in nematic elastomer membranes for certain classes of deformation. This idea has captured the interest of NASA as they seek lightweight and easily deployable space structures, and since the use of lightweight deployable membranes is often limited by wrinkling.

 

In order to understand the interplay of these non-linearities, we derive an elastic theory for nematic elastomers of small thickness. Our starting point is three-dimensional elasticity, and for this we incorporate the widely used model Bladon, Terentjev and Warner for the energy density of a nematic elastomer along with a Frank elastic penalty on nematic reorientation. We derive membrane and bending limits taking the thickness to zero by exploiting the mathematical framework of Gamma-convergence. This follows closely the seminal works of LeDret and Raoult on the membrane theory and Friesecke, James and Mueller on the bending theory.

 

Thu, 23 Oct 2014

12:00 - 13:00
L4

J.C. Maxwell's 1879 Paper on Thermal Transpiration and Its Relevance to Contemporary PDE

Marshall Slemrod
(University of Wisconsin - Madison)
Abstract
In his 1879 PRSL paper on thermal transpiration J.C.MAXWELL addressed the problem of steady flow of a dilute gas over a flat boundary. The experiments of KUNDT and WARBURG had demonstrated that if the boundary is heated with a temperature gradient , say increasing from left to right, the gas will flow from left to right. On the other hand MAXWELL using the continuum mechanics of his (and indeed our) day solved the ( compressible) NAVIER- STOKES- FOURIER equations for balance of mass, momentum, and energy and found a solution: the gas has velocity equal zero. The Japanese fluid mechanist Y. SONE has termed this the incompleteness of fluid mechanics. In this talk I will sketch MAXWELL's program and how it suggests KORTEWEG's 1904 theory of capillarity to be a reasonable “ completion” of fluid mechanics. Then to push matters in the analytical direction I will suggest that these results show that HILBERT's 1900 goal expressed in his 6th problem of passage from the BOLTZMANN equation to the EULER equations as the KNUDSEN number tends to zero in unattainable.
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