When we model natural phenomena, we usually need to describe how something changes in time depending on its position, for example how the current of a river evolves or how forest fires spread. The physical laws that govern these processes are formulated via so-called partial differential equations (PDEs). However, nature often is too complex to follow these equations alone and we model all the external effects, for example the influence of winds, by a random noise term. In my research, I study the resulting stochastic PDEs. In particular, my doctoral thesis focuses on understanding the probability of unlikely events, known as “large deviations”, for example the sudden change of velocity when the current encounters an obstacle.
Events like that form a part of our everyday experience because they happen at the visible level. The world around us, however, emerges from complex interactions between elementary particles, such as the strong force that binds the atomic nucleus together. The mathematical object used to describe these forces is called the Yang-Mills measure. In post-doctoral study, I aim to understand how stochastic PDEs can be used to describe that object in a toy model of our universe, where space-time is two-dimensional. Understanding the qualitative properties of this toy model will be a first step towards a rigorous mathematical framework for quantum mechanics, a long-standing open problem.