This is a project in mathematics, where we investigate loop groups – a form of symmetry that is ubiquitous in string theory and solid state theory. We investigate the way in which this specific form of symmetry can be represented in a quantum system (representations), and the way in which the associated conserved quantities are deformed in the transition from the classical to the quantum world (deformations).

Natural phenomena, like the weather, the flow of a river or a chemical reaction are modeled mathematically using equations involving differences. As an example, the wind is driven by differences in air pressure. Turbulence in these phenomena can be accounted for by adding a noise term to the equation. To study the properties of these equations, one needs to understand certain singularities related to this noise term. This project will study these singularities by decomposing them into simpler, bite-sized chunks. This will provide profound new insights on the solvability of these equations.

The voter model describes how individuals adopt opinions based on that of others. Of course, if individuals are far apart, they naturally are less likely to influence each other. In this research, we develop a mathematical framework in which we can incorporate this distance in the voter model in a general way. We are then interested in what its influence is on reaching consensus (i.e. everyone agreeing) and how long this takes. Furthermore, by looking backwards in time, we also study where opinions originate from in a network.

We investigate the motion of droplets on the nanometer scale. Here, the droplets particle nature comes into play, which leads to a statistical model for droplet motion by incorporating a noise term in the standard equation for the evolution. We will show that this model is mathematically sound and predicts that droplets spread faster on the nanometer scale compared to the standard deterministic model, in line with computer simulations.

Markov Chain Monte Carlo (MCMC) methods are clever computational techniques used to make predictions in uncertain situations - such as modeling disease spread, climate change, or decisions based on incomplete data. This workshop in Eindhoven brings together international experts and young researchers to explore new developments. MCMC enables smart sampling from millions of possibilities, making it feasible to use complex models. By improving these algorithms, scientists can draw more reliable conclusions from large and messy datasets.