L3

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.

We will discuss the percolation model on the hexagonal grid. In 2001 S. Smirnov proved conformal invariance of its scaling limit through the use of a tricky auxiliary combinatorial construction.

We present a more conceptual approach, implying that the construction in question can be thought of as geometrically natural one.

The main goal of the talk is to make it believable that not all nice and useful objects in the field have been already found.

No background is required.

We present sharp gradient estimates for the solution of the filtering equation and report on its applications in a high order cubature method for the nonlinear filtering problem.

I will present a mathematical model for the genetic evolution of a population which is divided in discrete colonies along a linear habitat, and for which the population size of each colony is random and constant in time. I will show that, under reasonable assumptions on the distribution of the population sizes, over large spatial and temporal scales, this population can be described by the solution to a stochastic partial differential equation with constant coefficients. These coefficients describe the effective diffusion rate of genes within the population and its effective population density, which are both different from the mean population density and the mean diffusion rate of genes at the microscopic scale. To do this, I will present a duality technique and a new convergence result for coalescing random walks in a random environment.

In this paper, we investigate the limit properties of frequency of empirical averages when random variables are described by a set of probability measures and obtain a law of large numbers for upper-lower probabilities. Our result is an extension of the classical Kinchin's law of large numbers, but the proof is totally different.

keywords: Law of large numbers,capacity, non-additive probability, sub-linear expectation, indepence

paper by: Zengjing Chen School of Mathematics, Shandong University and Qingyang Liu Center for Economic Research, Shandong University

Based on classical invariant theory, I describe a complete set of elements of the signature that is invariant to the general linear group, rotations or permutations.

A geometric interpretation of some of these invariants will be given.

Joint work with Jeremy Reizenstein (Warwick).