Given $d \ge 1$, $T>0$ and a vector field $\mathbf b \colon [0,T] \times \mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation $\partial_t u + \mathbf b \cdot \nabla u=0$ where $u \colon [0,T] \times \mathbb R^d \to \mathbb R$ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field $\mathbf b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost.
In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if $\mathbf b$ is locally of class $BV$ in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible $BV$ vector fields and thus gives a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini.
 

Let $f_1,\dots,f_k$ be real polynomials with no constant term and degree at most $d$. We will talk about work in progress showing that there are integers $n$ such that the fractional part of each of the $f_i(n)$ is very small, with the quantitative bound being essentially optimal in the $k$-aspect. This is based on the interplay between Fourier analysis, Diophantine approximation and the geometry of numbers. In particular, the key idea is to find strong additive structure in Fourier coefficients.

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications.

J. M. Foster 1 , N. E. Courtier 2 , S. E. J. O’Kane 3 , J. M. Cave 3 , R. Niemann 4 , N. Phung 5 , A. Abate 5 , P. J. Cameron 4 , A. B. Walker 3 & G. Richardson 2 .

1 School of Mathematics & Physics, University of Portsmouth, UK. {@email}

2 School of Mathematics, University of Southampton, UK.

3 School of Physics, University of Bath, UK.

4 School of Chemistry, University of Bath, UK.

5 Helmholtz-Zentrum Berlin, Germany.

Metal halide perovskite has emerged as a highly promising photovoltaic material. Perovskite-based solar cells now exhibit power conversion efficiencies exceeding 22%; higher than that of market-leading multi-crystalline silicon, and comparable to the Shockley-Queisser limit of around 33% (the maximum obtainable efficiency for a single junction solar cell). In addition to fast electronic phenomena, occurring on timescales of nanoseconds, they also exhibit much slower dynamics on the timescales of several seconds and up to a day. One well-documented example of this is the ‘anomalous’ hysteresis observed in current-voltage scans where the applied voltage is varied whilst the output current is measured. There is now a consensus that this is caused by the motion of ions in the perovskite material affecting the internal electric field and in turn the electronic transport.

We will discuss the formulation of a drift-diffusion model for the coupled electronic and ionic transport in a perovskite solar cell as well as its systematic simplification via the method of matched asymptotic expansions. We will use the resulting reduced model to give a cogent explanation for some experimental observations including, (i) the apparent disappearance of current-voltage hysteresis for certain device architectures, and (ii) the slow fading of performance under illumination during the day and subsequent recovery in the dark overnight. Finally, we suggest ways in which materials and geometry can be chosen to reduce charge carrier recombination and improve device performance.

Caustics are places where the light intensity diverges, and where the wave front has a singularity. We use a self-similar description to derive the detailed spatial structure of a cusp singularity, from where caustic lines originate. We also study singularities of higher order, which have their own, uniquely three-dimensional structure. We use this insight to study shock formation in classical compressible Euler dynamics. The spatial structure of these shocks is that of a caustic, and is described by the same similarity equation.

Many types of patterns emerging spontaneously can be observed in systems involving thin elastic plates and subjected to external or internal stresses (compression, differential growth, shearing, tearing, etc.). These mechanical systems can sometime be seen as model systems for more complex natural systems and allow to study in detail elementary emerging patterns. One of the simplest among such systems is a bilayer composed of a thin plate resting on a thick deformable substrate. Upon slight compression, periodic undulations (wrinkles) with a well-defined wavelength emerge at the level of the thin layer. We will show that, as the compression increases, this periodic state is unstable and that a second order transition to a localized state (fold) occurs when the substrate is a dense fluid.

The desire to “freely suspend the constituents of matter” in order to study their behavior can be traced back over 200 years to the diaries of Lichtenberg. From radio-frequency ion traps to optical tweezing of colloidal particles, existing methods to trap matter in free space or solution rely on the use of external fields that often strongly perturb the integrity of a macromolecule in solution. We recently introduced the ‘electrostatic fluidic trap’, an approach that exploits equilibrium thermodynamics to realise stable, non-destructive confinement of single macromolecules in room temperature fluids, and represents a paradigm shift in a nearly century-old field. The spatio-temporal dynamics of a single electrostatically trapped object reveals fundamental information on its properties, e.g., size and electrical charge. We have demonstrated the ability to measure the electrical charge of a single macromolecule in solution with a precision much better than a single elementary charge. Since the electrical charge of a macromolecule in solution is in turn a strong function of its 3D conformation, our approach enables for the first time precise, general measurements of the relationship between 3D structure and electrical charge of a single macromolecule, in real time. I will present our most recent advances in this emerging area of molecular measurement and show how such high-precision measurement at the nanoscale may be able to unveil the presence of previously unexpected phenomena in intermolecular interactions in solution.