We will discuss the solution of eigenvalue problems associated with partial differential equations (PDE)s that can be written in the generalised form Ax = λMx, where the matrices A and/or M may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilised formulations are used for the numerical approximation of PDEs. With the help of classical numerical examples we will show that the presence of one (or both) parameters can produce unexpected results.

Hessian-dependent functionals play a pivotal role in a wide latitude of problems in mathematics. Arising in the context of differential geometry and probability theory, this class of problems find applications in the mechanics of deformable media (mostly in elasticity theory) and the modelling of slow viscous fluids. We study such functionals from three distinct perspectives.

We present a theoretical analysis of the Weak Adversarial Networks (WAN) method, recently proposed in [1, 2], as a method for approximating the solution of partial differential equations in high dimensions and tested in the framework of inverse problems. In a very general abstract framework.

Analogies between financial markets and critical phenomena have long been observed empirically. So far, no convincing theory has emerged that can explain these empirical observations. Here, we take a step towards such a theory by modeling financial markets as a lattice gas.

Wave propagation and scattering problems are of huge importance in many applications in science and engineering - e.g., in seismic and medical imaging and more generally in acoustics and electromagnetics.

In this short course, we will introduce some elements in deriving the hp a posteriori error estimate for a high-order unfitted finite element method for elliptic interface problems. The key ingredient is an hp domain inverse estimate, which allows us to prove a sharp lower bound of the hp a posteriori error estimator.

Random fields are popular models in statistics and machine learning for spatially dependent data on Euclidian domains. However, in many applications, data is observed on non-Euclidian domains such as street networks. In this case, it is much more difficult to construct valid random field models. In this talk, we discuss some recent approaches to modeling data in this setting, and in particular define a new class of Gaussian processes on compact metric graphs.

In this talk, we begin by a brief introduction to proper scoring rules and their use in statistics. Then, we discuss an often overlooked problem: the up-weighting of observations with large uncertainty, which can lead to unintuitive rankings of models, by some of the most popular proper scoring rules, such as the continuously ranked probability score (CRPS), the MAE, and the MSE.