The research proposed lies at the interface of dynamical systems and several other mathematical disciplines, and in particular number theory and the theory of automorphic form. It is well known that the collections of invariant probability measures and closed invariant sets for hyperbolic maps or flows is very large; remarkably, in many dynamical systems of algebraic origin where there are two (or more) commuting hyperbolic maps or flows it is conjectured that there are actually very few measures or closed sets invariant under this bigger action. Substantial progress has been made in the study of such systems, which has yielded a proof of arithmetic quantum unique ergodicity of compact arithmetic surfaces, and has given a partial result towards Littlewood's Conjecture in diophantine approximation. The author proposes to build on his research on the various facets of this problem to further our understanding of this rigidity phenomenon, as well as to apply these techniques and methods and the methods developed by other authors towards potential applications in number theory and other subjects.
In dynamical systems we study evolution of a mathematical system. Ergodic theory is a specific flavor of the theory of dynamical systems. The tools of ergodic theory applied to concrete algebraically defined dynamical systems can be used to prove theorems and conjectures in other seemingly unrelated fields, notably number theory. In our previous work we have shown how these tools can also be used effectively to study problems motivated by quantum mechanics, specifically the arithmetic case of the Quantum Unique Ergodicity Conjecture. The connections that have been found between these special dynamical systems and problems in number theory, mathematical physics, and automorphic forms (the study of the spectrum of certain fundamental arithmetic manifolds) suggests that this interdisciplinary approach is likely to have additional applications. In addition to the research itself, substantial effort will be given to human resources development, particularly at the graduate level.
