University of Warwick

The Cohen-Lenstra heuristics, postulated in the early 80s, conceptually explained numerous phenomena in the behaviour of ideal class groups of number fields that had puzzled mathematicians for decades, by proposing a probabilistic model: the probability that the class group of an imaginary quadratic field is isomorphic to a given group $A$ is inverse proportional to $\#\text{Aut}(A)$. This is a very natural model for random algebraic objects. But the probability weights for more general number fields, while agreeing well with experiments, look rather mysterious. I will explain how to recover the original heuristic in a very conceptual way by phrasing it in terms of Arakelov class groups instead. The main difficulty that one needs to overcome is that Arakelov class groups typically have infinitely many automorphisms. We build up a theory of commensurability of modules, of groups, and of rings, in order to remove this obstacle. This is joint work with Hendrik Lenstra.

I will describe joint work with Manjul Bhargava (Princeton) and Tom Fisher (Cambridge) in which we determine the probability that random equation from certain families  has a solution either locally (over the reals or the p-adics), everywhere locally,  or globally. Three kinds of equation will be considered: quadratics in any number of variables, ternary cubics and hyperelliptic quartics.

Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until recently.

The theory of regularity structures allows one to give a meaning to several stochastic PDEs, including the Parabolic Anderson Model. So far, these equations have been considered on a torus. The goal of this talk is to explain how one can define the PAM on the whole space R^3. This is a joint work with Martin Hairer.

When solving Einstein's equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike (or null) hypersurface. At the PDE level, one finds that the boundary data is typically prescribed on a surface at which the equations become singular and standard energy estimates break down. I will discuss how to handle this singularity by introducing a renormalisation procedure. I will also talk about the consequences of different choices of boundary conditions for solutions of Einstein’s equations with negative cosmological constant.

In this talk I will present several results connected with the idea of magnetic relaxation for MHD, including some new commutator estimates (and a counterexample to the estimate in the critical case). (Joint work with various subsets of  D. McCormick, J. Robinson, C. Fefferman and J-Y. Chemin.)

Abstract: The signature of a path characterizes the non-commutative evolvements along the path trajectory. Nevertheless, one can extract local commutativities from the signature, thus leading to an inversion scheme.

We consider a large class of three dimensional continuous dynamic fluctuation models, and show that they all rescale and converge to the stochastic Allen-Cahn equation, whose solution should be interpreted after a suitable renormalization procedure. The interesting feature is that, the coefficient of the limiting equation is different from one's naive guess, and the renormalization required to get the correct limit is also different from what one would naturally expect. I will also briefly explain how the recent theory of regularity structures enables one to prove such results. Joint work with Martin Hairer.

For many questions of scientific interest, all-atom molecular simulations are still out of reach, in particular in materials engineering where large numbers of atoms, and often expensive force fields, are required. A long standing challenge has been to construct concurrent atomistic/continuum coupling schemes that use atomistic models in regions of space where high accuracy is required, with computationally efficient continuum models (e.g., FEM) of the elastic far-fields.

Many different mechanisms for achieving such atomistic/continuum couplings have been developed. To assess their relative merits, in particular accuracy/cost ratio, I will present a numerical analysis framework. I will use this framework to analyse in more detail a particularly popular scheme (the BQCE scheme), identifying key approximation parameters which can then be balanced (in a surprising way) to obtain an optimised formulation.

Finally, I will demonstrate that this analysis shows how to remove a severe bottlenecks in the BQCE scheme, leading to a new scheme with optimal convergence rate.