Evolution equations in spaces of measures describe a wide variety of natural phenomena. The theory for such evolutions has seen tremendous growth in the last decades, of which resulted in AGS and RI theory for analysing variational evolutions—evolutions driven by one or more energies/entropies. The AGS theory provides a rich framework to study gradient flows in general metric spaces, where the Wasserstein metric of optimal transport theory plays a fundamental role in the case of probability measures. While AGS and RI theory has allowed massive development of variational evolutions in a certain direction—gradient flows with homogeneous dissipation—physics and large-deviation theory suggest the study of generalised gradient flows—gradient flows with non-homogeneous dissipation—which are not covered in either theories. On the other hand, numerical analysis for measure-valued evolutions is lagging behind. Most numerical schemes are not well-suited for such evolutions because they were not developed with measures in mind. In this proposal, we remedy these deficiencies by introducing dynamical-variational transport costs (DVTs), a class of large-deviation inspired functionals that provide a variational generalisation of several existing transport distances. Since DVTs generate non-homogeneous generalisations of length spaces and are stable under weak convergence, they will be used to extend metric-space techniques to general length spaces, thereby allowing a variational framework for (1) generalised gradient flows to be rigorously investigated, and (2) the multiscale analysis of such evolutions used in the development of numerical schemes. An in-depth investigation of well-posedness for DVTs and their stability under parameter perturbation will not only give rise to a new theory for variational evolutions on non-metric spaces, where the interplay between large-deviation and optimal transport theory is made explicit, but also provides new numerical schemes that are specifically designed to handle gradient flows in spaces of measures.

Quantum computers are made up of many entities that, not only interact with each other but also with the outside world. These interactions give quantum computers the power they need to surpass classical computers, but at the same time make them difficult to control due to a phenomenon called quantum decoherence, or in other words, noise. In this project, researchers develop noise-resistant algorithms, paving the way for currently available quantum computers to solve hard problems in cryptography, and to study chemical processes for the development of new materials and medicine more effectively than currently possible.

All real-world systems are made up of many entities that constantly interact with each other. Controlling such systems, thus, amounts to controlling every entity within the system—an unmanageable task for large interacting systems in, e.g., cellular biology and transport networks. In this project, researchers develop a mathematical theory to optimally control effective systems that correspond to the limit of very large interacting systems. The resulting theory enables the development of algorithms that could potentially facilitate the design of, i.a., synthetic biological robots and intelligent transport systems.

In the present project we aim at investigating the dynamics of particles under non-ideal turbulent flow conditions. We aim at a considerable step forward with respect to past investigations that focused mostly on homogenous and isotropic turbulence. We plan to move in the direction of geophysical fluid dynamics investigating how the dynamics of particles is influenced by the presence of shear and by thermal convective motions. Both shear and convective motions are ordinarily found in atmospheric flows and their interplay is fundamental, for example, for cloud microphysics.

This project is set out to digitize the control loop of switched-mode systems for use in power driving electronics. This offers opportunities to use more complex control algorithms which can improve power efficiency and reduce distortion compared to analog techniques. The results are aimed at applications like class D amplifiers and motor controllers.