L4

This talk is about a 4d "(ambi-)twistor string" formula for sEYM tree-level scattering amplitudes and is work in progress. The formula sheds some new light on the translation between CHY formulae and (ambi-)twistor formulae in general, potentially even at loop level.

The theory of connections on curves and Hitchin systems is something like a “global theory of Lie groups”, where one works over a Riemann surface rather than just at a point. We’ll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case), and discuss their classification, outlining a theory of “Dynkin diagrams” as a step towards classifying some examples of such objects.

We determine the hydrodynamic limit of some kinetic equations with either stochastic Vlasov force term or stochastic collision kernel. We obtain stochastic second-order parabolic equations at the limit. In the regime we consider, we also observe (or do not observe) some phenomena of enhanced diffusion. Joint works with Nils Caillerie, Arnaud Debussche, Martina Hofmanová.
 

We will give an overview of the Soliton Resolution Conjecture, focusing mainly on the Wave Maps Equation. This is a program about understanding the formation of singularities for a variety of critical hyperbolic/dispersive equations, and stands as a remarkable topic of research in modern PDE theory and Mathematical Physics. We will be presenting our contributions to this field, elaborating on the required background, as well as discussing some of the latest results by various authors.

Conventional neutron reflection is a very powerful tool to characterise surfactants, polymers and other materials at the solid/liquid and air/liquid interfaces. Usually the analysis considers molecular layers with coherent addition of reflected waves that give the resultant reflected intensity. In this short workshop talk I will illustrate recent developments in this approach to address a wide variety of challenges of academic and commercial interest. Specifically I will introduce the challenges of using substrates that are thick on the coherence lengthscale of the radiation and the issues that brings in the structural analysis. I also invite the audience to consider if there may be some mathematical analysis that might lead us to exploit this incoherence to optimise our analysis. In particular, facilitating the removal of the 'background substrate contribution' to help us focus on the adsorbed layers of most interest.

We describe a convergence acceleration technique for generic optimization problems. Our scheme computes estimates of the optimum from a nonlinear average of the iterates produced by any optimization method. The weights in this average are computed via a simple linear system, whose solution can be updated online. This acceleration scheme runs in parallel to the base algorithm, providing improved estimates of the solution on the fly, while the original optimization method is running. Numerical experiments are detailed on classical classification problems.
 

Results on the existence and stability of clustered spike patterns for biological reaction‐diffusion systems with two small diffusivities will be presented. In particular we consider a consumer chain model and the Gierer‐Meinhardt activator-inhibitor system with a precursor gradient. A clustered spike pattern consists of multiple spikes which all approach the same limiting point as the diffusivities tend to zero. We will present results on the asymptotic behaviour of the spikes including their shapes, positions and amplitudes. We will also compute the asymptotic behaviour of the eigenvalues of the system linearised around a clustered spike pattern. These systems and their solutions play an important role in biological modelling to account for the bridging of lengthscales, e.g. between genetic, nuclear, intra‐cellular, cellular and tissue levels, or for the time-hierarchy of biological processes, e.g. a large‐scale structure, which appears first, induces patterns on smaller scales. This is joint work with Juncheng Wei.
 

In everyday language, this talk studies the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and for short times we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the well-posedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using a new decomposition of $L^2(\Rd)$ into heat packets from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated numerically by solving a sequence of finite-dimensional optimization problems. (joint with Alden Waters)
 

I will report some recent analytical results on microstructures in low-hysteresis shape memory alloys. The modelling assumption is that the width of the thermal hysteresis is closely related to the minimal energy that is necessary to build a martensitic nucleus in an austenitic matrix. This energy barrier is typically modeled by (singularly perturbed) nonconvex elasticity functionals. In this talk, I will discuss recent results on the resulting variational problems, including stress-free inclusions and microstructures in the case of almost compatible phases. This talk is partly based on joint works with S. Conti, J. Diermeier, M. Klar, and D. Melching.