In 1973, Serre defined $p$-adic modular forms as limits of modular forms, and constructed the Leopoldt-Kubota $L$-function as the constant term of a limit of Eisenstein series. This was extended by Deligne-Ribet to totally real number fields, and Lauder and Vonk have developed an algorithm for interpolating $p$-adic $L$-functions of such fields using Serre's idea. We explain what an $L$-function is and why you should care, and then move on to giving an overview of the algorithm, extensions, and applications.

In this leisure talk I will show how a sum of squares decomposition problem can be transformed to a problem of semi-definite optimization. Then the practicality of such reformulations will be discussed, illustrated by an explicit example of Artin's solutions to Hilberts 17th problem. Finally I will show how a numerical solution could be turned into a mathematically certified one, using the order structure on the cone of sums of squares.
The talk requires no pre-requisite knowledge of neither optimization or programming and only undergraduate mathematics.

We consider the problem of parameter estimation for a McKean stochastic differential equation, and the associated system of weakly interacting particles. The problem is motivated by many applications in areas such as neuroscience, social sciences (opinion dynamics, cooperative behaviours), financial mathematics, statistical physics. We will first survey some model properties related to propagation of chaos and ergodicity and then move on to discuss the problem of parameter estimation both in offline and on-line settings. In the on-line case, we propose an online estimator, which evolves according to a continuous-time stochastic gradient descent algorithm on the asymptotic log-likelihood of the interacting particle system. The talk will present our convergence results and then show some numerical results for two examples, a linear mean field model and a stochastic opinion dynamics model. This is joint work with Louis Sharrock, Panos Parpas and Greg Pavliotis. Preprint: https://arxiv.org/abs/2106.13751

We study sampling from a target distribution $e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm. For any potential function $f$ whose tails behave like $\|x\|^\alpha$ for $\alpha \in [1,2]$, and has $\beta$-H\"older continuous gradient, we derive the sufficient number of steps to reach the $\epsilon$-neighborhood of a $d$-dimensional target distribution as a function of $\alpha$ and $\beta$. Our rate estimate, in terms of $\epsilon$ dependency, is not directly influenced by the tail growth rate $\alpha$ of the potential function as long as its growth is at least linear, and it only relies on the order of smoothness $\beta$.

Our rate recovers the best known rate which was established for strongly convex potentials with Lipschitz gradient in terms of $\epsilon$ dependency, but we show that the same rate is achievable for a wider class of potentials that are degenerately convex at infinity.

The success of operator splitting techniques for convex optimization has led to an explosion of methods for solving large-scale and non convex optimization problems via convex relaxation. 

This success is at the cost of overlooking direct approaches to operator splitting that embrace some of the more inconvenient aspects of many model problems, namely nonconvexity, non smoothness and infeasibility.  I will introduce some of the tools we have developed for handling these issues, and present sketches of the basic results we can obtain.

The formalism is in general metric spaces, but most applications have their basis in Euclidean spaces.  Along the way I will try to point out connections to other areas of intense interest, such as optimal mass transport.

The aim of this talk is to understand the qualitative properties that emerge from a PDE model inspired from neurosciences, in order to understand what are the key processes that lead to mathematical complex patterns for the solutions of this equation. 

In this talk I will discuss the development and justification of linearised shock-capturing for aeronautical applications such as flutter, forced response and design optimisation.  At its core is a double-limiting process, reducing both the viscosity and the size of the unsteady or steady perturbation to zero. The design optimisation also requires the consideration of the adjoint equations, but with shock-capturing this is best done at the level of the numerical discretisation, rather than the PDE.

We  report  on the theory of evolution equations that combine a strongly nonlinear parabolic character with the presence of fractional operators representing long-range interaction effects, mainly of fractional Laplacian type. Examples include nonlocal porous media equations and fractional p-Laplacian operators appearing in a number of variants. 

Recent work concerns the time-dependent fractional p-Laplacian equation with parameter p>1 and fractional exponent 0<s<1. It is the gradient flow corresponding to the Gagliardo–Slobodeckii fractional energy. Our main interest is the asymptotic behavior of solutions posed in the whole Euclidean space, which is given by a kind of Barenblatt solution whose existence relies on a delicate analysis. The superlinear and sublinear ranges involve different analysis and results. 
 

Group theory is littered with undecidable problems. A classic example is the word problem: there are groups for which there exists no algorithm that can decide if a product of generators represents the trivial element or not. Many problems (the word problem included) are at least semidecidable, meaning that there is a correct algorithm guaranteed to terminate if the answer is "yes", but with no guarantee on how long one has to wait. I will discuss strategies to try and tackle various semidecidable problems computationally using modern solvers for Boolean satisfiability, with the key example being the discovery of a counterexample to the Kaplansky unit conjecture.