Institute for Probability, Analysis and Dynamics

Research on various aspects of probability theory, the science of uncertainty around us, leads to frontier-line results concerning applications of information-theoretic methods to prove fundamental theorems in probability, and limit theorems on stochastic processes that originate from statistical physics, earth sciences, biology, computer science or combinatorial structures like random graphs and networks, interacting particle systems or processes with long memory. We investigate many such models and, besides advancing probability itself, our results find applications all across mathematics as well as in other sciences listed above and beyond.

Analysis is a diverse field that explores the fundamental properties of continuous and discrete systems. Key research areas in Bristol include Geometric Spectral Theory: investigating the relationship between the sound of a drum reflecting the spectrum of self-adjoint operators, such as the Laplacian and the shape of the drum translating into the geometry of manifolds or domains in Euclidean space. For typical hyperbolic surfaces, Mirzakhani’s innovative probabilistic methods provide a unique approach to studying their spectrum. Geometric Function Theory: examining how infinitesimal properties of functions, like conformality, lead to significant large-scale geometric consequences. Harmonic Analysis: studying operators acting on sequence-space functions in the discrete setting, the oscillatory singular integrals, and pointwise convergence in Euclidean spaces. Other research areas include studying weighted isoperimetric inequality in Euclidean space closely related to Geometric Spectral Theory.