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.

The development of advanced vision-language models necessitates considerable resources, both in terms of computation and data. There is growing interest in training these models efficiently and effectively and leveraging them for various downstream tasks. This dissertation presents several contributions aimed at improving both learning and data efficiency in vision-language learning, and how to leverage them into downstream tasks.

Sequential modeling algorithms have made significant strides in a variety of domains, facilitating intelligent decision-making and planning in complex scenarios. This dissertation explores the potential and limitations of these algorithms, unveiling novel approaches to enhance their performance across diverse fields, from autonomous driving and trajectory forecasting to reinforcement learning and vision language understanding.

In computer vision, generative AI models are typically built for images, videos, and 3D objects. Recently, there has emerged a paradigm of neural fields, which unifies the representations of such types of data by parametrizing them via neural networks. In this thesis, we develop generative models for images, videos, and 3D scenes which treat the underlying data in such a form and explore the benefits which such a perspective provides.

Datasets that capture the connection between vision, language, and affection are limited, causing a lack of understanding of the emotional aspect of human intelligence. As a step in this direction, the ArtEmis dataset was recently introduced as a large-scale dataset of emotional reactions to images along with language explanations of these chosen emotions.

We present a novel large-scale dataset and accompanying machine learning models aimed at providing a detailed understanding of the interplay between visual content, its emotional effect, and explanations for the latter in language. In contrast to most existing annotation datasets in computer vision, we focus on the affective experience triggered by visual artworks and ask the annotators to indicate the dominant emotion they feel for a given image and, crucially, to also provide a grounded verbal explanation for their emotion choice.