Jones will study almost everywhere convergence in analysis and ergodic theory. He will investigate questions concerning almost everywhere convergence for absolutely integrable functions, the central limit theorem on the real line and its transference to the ergodic setting, and questions related to the solution of an old problem of Hopf. Singular integral techniques will be further developed and applied. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.
