A main strand of research in the Analysis group is the interplay between the spectrum of the Laplace operator with self-adjoint boundary conditions and the geometry of the manifold or domain in Euclidean space. Various tools such as the theory of function spaces, calculus of variations and shape optimization techniques play a key role. A closely related area is the stochastic analysis of Brownian motion, and quantitative properties of solutions of the heat equation both on Riemannian manifolds and in domains in Euclidean space. Other research themes in the group include the theory of function spaces and geometric function theory. Geometric function theory is concerned with how infinitesimal properties of functions, such as conformality, have large-scale geometric consequences. All these topics have links to many other groups at Bristol, such as Ergodic Theory and Dynamical Systems, Mathematical Physics, and Group Theory.