This talk is concerned with quantitative periodic homogenization in domains with boundaries. The quantitative analysis near boundaries leads to the study of boundary layers correctors, which have in general a nonperiodic structure. The interaction between the boundary and the microstructure creates geometric resonances, making the study of the asymptotics or continuity properties particularly challenging. The talk is based on work with S. Armstrong, T. Kuusi and J.-C. Mourrat, as well as work by Z. Shen and J. Zhuge

In this talk I will start with a brief overview of the Cauchy problem for the Einstein equations of general relativity, and in particular the nonlinear stability of the trivial Minkowski solution in wave gauge as shown by Lindblad and Rodnianski. I will then discuss the Kaluza Klein spacetime of the form $R^{1+3} \times K$ where $K$ is the $n-$torus with the flat metric.  An interesting question to ask is whether this solution to the Einstein equations, viewed as an initial value problem, is stable to small perturbations of the initial data. Motivated by this problem, I will outline how the proof of stability in a restricted class of perturbations in fact follows from the work of Lindblad and Rodnianski, and discuss the physical justification behind this restriction. 

The Fujita equation $u_{t}=\Delta u+u^{p}$, $p>1$, has been a canonical blow-up model for more than half a century. A great deal is known about the singularity formation under a variety of conditions. In particular we know that blow-up behaviour falls broadly into two categories, namely Type I and Type II. The former is generic and stable while the latter is rare and highly unstable. One of the central results in the field states that in the Sobolev subcritical regime, $1<p<\frac{n+2}{n-2}$, $n\geq 3$, only type I is possible whenever the domain is \emph{convex} in $\mathbb{R}^n$. Despite considerable effort the requirement of convexity has not been lifted and it is not clear whether this is an artefact of the methodology or whether the geometry of the domain may actually affect the blow-up type. In my talk I will discuss how the question of the blow-up type for non-convex domains is intimately related to the validity of some Li-Yau-Hamilton inequalities.

Phase transitions and critical phenomena are ubiquitous in Nature. They permeate physics, chemistry, biology and complex systems in general, and are characterized by the role of correlations and fluctuations of many degrees of freedom. From a mathematical viewpoint, in the vicinity of a critical point, thermodynamic quantities exhibit singularities and scaling properties. Theoretical attempts to describe classical phase transitions using tools from differential topology and Morse theory provided strong arguments pointing that a phase transition may emerge as a consequence of topological changes in the configuration space around the critical point.

On the other hand, much work was done concerning the topology of networks which spontaneously emerge in complex systems, as is the case of the genome, brain, and social networks, most of these built intrinsically based on measurements of the correlations among the constituents of the system.

We aim to transpose the topological methodology previously applied in n-dimensional manifolds, to describe phenomena that emerge from correlations in a complex system, in which case Hamiltonian models are hard to invoke. The main idea is to embed the network onto an n-dimensional manifold and to study the equivalent to level sets of the network according to a filtration parameter, which can be the probability for a random graph or even correlations from fMRI measurements as height function in the context of Morse theory.  By doing so, we were able to find topological phase transitions either in random networks and fMRI brain networks.  Moreover, we could identify high-dimensional structures, in corroboration with the recent finding from the blue brain project, where neurons could form structures up to eleven dimensions.The efficiency and generality of our methodology are illustrated for a random graph, where its Euler characteristic can be computed analytically, and for brain networks available in the human connectome project.  Our results give strong arguments that the Euler characteristic, together with the distributions of the high dimensional cliques have potential use as topological biomarkers to classify brain Networks. The above ideas may pave the way to describe topological phase transitions in complex systems emerging from correlation data.

A granular material forms a distinct and fascinating phase in physics -- sand acts as a fluid as grains flow through your fingers, the fallen grains form a solid heap on the floor or may suspend in the wind like a gas.

The main challenge of studying granular materials is the development of constitutive models valid across scales, from the micro-scale (collisions between individual particles), via the meso-scale (flow structures inside avalanches) to the macro-scale (dunes, heaps, chute flows).

In this talk, I am highlighting three recent projects from my laboratory, each highlighting physical behavior at a different scale. First, using the property of birefringence, we are quantifying both kinetic and dynamic properties in an avalanche of macroscopic particles and measure rheological properties. Secondly, we explore avalanches on an erodible bed that display an intriguing dynamic intermittency between regimes. Lastly, we take a closer look at aqueous (water-driven) dunes in a novel rotating experiment and resolve an outstanding scaling controversy between migration velocity and dune dimension.

A Surstseyan eruption is a particular kind of volcanic eruption which involves the bulk interaction of water and hot magma. Surtsey Island was born during such an eruption process in the 1940s. I will talk about mathematical modelling of the flashing of water to steam inside a hot erupted lava ball called a Surtseyan bomb. The overall motivation is to understand what determines whether such a bomb will fragment or just quietly fizzle out...
Partial differential equations model transient changes in temperature and pressure in Surtseyan ejecta. We have used a highly simplified approach to the temperature behaviour, to separate temperature from pressure. The resulting pressure diffusion equation was solved numerically and asymptotically to derive a single parametric condition for rupture of ejecta. We found that provided the permeability of the magma ball is relatively large, steam escapes rapidly enough to relieve the high pressure developed at the flashing front, so that rupture does not occur. This rupture criterion is consistent with existing field estimates of the permeability of intact Surtseyan bombs, fizzlers that have survived.
I describe an improvement of this model that allows for the fact that pressure and temperature are in fact coupled, and that the process is not adiabatic. A more systematic reduction of the resulting coupled nonlinear partial differential equations that arise from mass, momentum and energy conservation is described. We adapt an energy equation presented in G.K. Batchelor's book {\em An Introduction to Fluid Dynamics} that allows for pressure-work. This is work in progress.  Work done with Emma Greenbank, Ian Schipper and Andrew Fowler 

This talk will be a general introduction to perverse sheaves and their applications to the study of algebraic varieties, with a view towards enumerative geometry. It is aimed at non-experts.

We will start by considering constructible sheaves and local systems, and how they relate to the notion of stratification: this offers some insight in the relationship with intersection cohomology, which perverse sheaves generalise in a precise sense.

We will then introduce some technical notions, like t-structures, perversities, and intermediate extensions, in order to define perverse sheaves and explore their properties.

Time permitting, we will consider the relevant example of nearby and vanishing cycle functors associated with a critical locus, their relationship with the (hyper)-cohomology of the Milnor fibre and how this is exploited to define refined enumerative invariants in Donaldson-Thomas theory.