OxPDE, University of Oxford

we study a new mechanism of reaction-diffusion involving a line with fast diffusion, proposed to model the influence of transportation networks on biological invasions. 
We will be interested in the existence and uniqueness of traveling waves solutions, and especially focus on their velocity. We will show that it grows as the square root of the diffusivity on the line, generalizing and showing the robustness of a result by Berestycki, Roquejoffre and Rossi (2013), and provide a characterization of the growth ratio thanks to an hypoelliptic (a priori) degenerate system. 
Finally we will take a look at the dynamics and show that the waves attract a large class of initial data. In particular, we will shed light on a new mechanism of attraction which enables the waves to attract initial data with size independent of the diffusion on the line : this is a new result, in the sense than usually, enhancement of propagation has to be paid by strengthening the assumptions on the size of the initial data for invasion to happen.

For maps from surfaces there is a close connection between the area of the surface parametrised by the map and its Dirichlet energy and this translates also into a relation for the corresponding critical points. As such, when trying to find minimal surfaces, one route to take is to follow a suitable gradient flow of the Dirichlet energy. In this talk I will introduce such a flow which evolves both a map and a metric on the domain in a way that is designed to change the initial data into a minimal immersions and discuss some question concerning the existence of solutions and their asymptotic behaviour. This is joint work with Peter Topping.

TBA

Dislocations are line defects in crystals, and were first posited as the carriers of plastic flow in crystals in the 1934 papers of Orowan, Polanyi and Taylor. Their hypothesis has since been experimentally verified, but many details of their behaviour remain unknown. In this talk, I present joint work with Christoph Ortner on an infinite lattice model in which screw dislocations are free to be created and annihilated. We show that configurations containing single geometrically necessary dislocations exist as global minimisers of a variational problem, and hence are globally stable equilibria amongst all finite energy perturbations.

Penrose’s Weyl Curvature Hypothesis, which dates from the late 70s, is a hypothesis, motivated by observation, about the nature of the Big Bang as a singularity of the space-time manifold. His Conformal Cyclic Cosmology is a remarkable suggestion, made a few years ago and still being explored, about the nature of the universe, in the light of the current consensus among cosmologists that there is a positive cosmological constant.  I shall review both sets of ideas within the framework of general relativity, and emphasise how the second set solves a problem posed by the first.

We consider minimisers of integral functionals $F$ over a ‘constrained’ class of $W^{1,p}$-mappings from a bounded domain into a compact Riemannian manifold $M$, i.e. minimisers of $F$ subject to holonomic constraints. Integrands of the form $f(Du)$ and the general $f(x,u,Du)$ are considered under natural strict $p$-quasiconvexity and $p$-growth assumptions for any exponent $1 < p <+\infty$. Unlike the harmonic and $p$-harmonic map case, the quasiconvexity condition requires one to linearise the map at the level of the gradient. In a bid to give a direct proof of partial $C^{1,α}-regularity for such minimisers, we developing an appropriate notion of a tangential harmonic approximation together with a discussion on the difficulties in establishing Caccioppoli-type inequalities. The need in the latter problem to construct suitable competitors to the minimiser via the so-called Luckhaus Lemma presents difficulties quite separate to that of the unconstrained case. We will give a proof of this lemma together with a discussion on the implications for higher integrability.

I will report my work on rough solutions to Cauchy problem for the Einstein vacuum equations in CMC spacial harmonic gauge, in which we obtain the local well-posedness result in $H^s$, $s$&gt;$2$. The novelty of this approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric $\bf{g}$, we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation $\Box_{\bf{g} } \phi=0$ directly. If time allows, I will talk about my work in progress on the sharp results for the more general quasilinear wave equations by vector fields approach.