Analysis

The analysis group at Durham has a broad of range of research interests. The unifying theme among the members in this group is the use of analytic tools to study topics in analysis and its interactions with other areas of mathematics. Here is a short overview of some of the main directions of research of the group members.

Analytic number theory

Analytic number theory is a branch of number theory which uses techniques from analysis to answer number theoretic questions ranging from distribution of primes, integer solutions of Diophantine equations to modular forms and L-functions.

Ergodic Theory and Dynamical Systems

The field of dynamical systems and ergodic theory has wide-ranging applications in mathematics, from number theory to geometry and topology. There are many problems in mathematics which, though not having an obvious temporal component, can be seen through the lens of a dynamical system. In its most basic form, a dynamical system is just a process governing how an object develops over time, often preserving some quantity such as volume or measure. As even the most simple seeming rule (such as angle doubling on the circle) can be chaotic, Ergodic Theory is a way to derive a lot of information about how a system evolves, in a statistical sense (what happens “typically”). Members of the analysis group are actively involved in the ergodic theory of shift spaces, group actions, flows on manifolds, fractal geometry, geometric group theory, and connections of these topics with problems in number theory and spectral theory.

Partial Differential Equations

Starting with the works of Euler on the motion of perfect fluids almost 300 years ago, Partial Differential Equations (PDEs) have been an essential and fundamental tool in the study of complex phenomena including particles dynamics, chemical reactions, biological systems, weather prediction, economic modelling, social interactions, machine learning, and many others.
The research interests of members of our group include fluid mechanics, kinetic theory, mean field games, mean field limits, and various models of physical, biological, and chemical phenomena. Besides tools from analysis, we often also use ideas and tools from probability, geometry, dynamical systems, spectral theory, and the modern theory of optimal transportation.

Spectral Theory

Spectral Theory is the study of operators on infinite dimensional vector spaces. It is an interesting topic in its own right and has a wealth of applications across analysis, for example to PDEs. It also enjoys a rich interplay with other fields including geometry, mathematical physics, probability and beyond. Members of the analysis group investigate the spectra of operators and explore links between spectral concepts (such as spectral gap, spectral enclosures, nodal domains) and the aforementioned fields.