Applied Project --- The Birkhoff Ergodic Theorem
The Birkhoff Ergodic Theorem forms the foundation of the field of ergodic theory. In its most austere form it answers the following question:
Take an infinite sequence S_n of numbers, and a real-valued function f. Compute the average of f evaluated along the sequence S_n. Does the limit of the average as n tends to infinity exist? And if so, what is the limit?
For a very wide class of sequences S_n and functions f, the answers are "yes" and "to the integral of f" - simple answers which give rise to a powerful mathematical framework. The ideas behind this theorem originated in statistical physics (Boltzmann's Ergodic Hypothesis), but it has since proved valuable in a wide variety of contexts, including dynamical systems, number theory (Gelfand's problem), statistics (the Strong Law of Large Numbers), classical mechanics and chaos theory.
This project will be largely investigatory and analytical, but there is also some scope for numerical work. Interested candidates should contact Dr Rob Sturman.