The principal investigator will work on problems from the theory of dynamical systems related to Diophantine properties of real numbers. The first part of the project involves the development of combinatorial models for semisimple flows that generalize the symbolic description of the geodesic flow on the modular surface given by continued fractions. These models will provide important tools for addressing several open problems in Diophantine approximation, including the famous Littlewood Conjecture, Schmidt's Conjecture (on successive minima of a lattice), and determining the Hausdorff dimension of the set of singular vectors. This part of the project may also involve some numerical studies. The second part of the project focuses on problems related to the ergodic theory of rational billiards. One objective will be to test the validity of the conjectural picture that every nonergodic Teichmuller geodesic arises from a Masur-Smillie-type construction. Another objective is to understand how the Hausdorff dimension of the set of nonergodic directions depends on the Diophantine properties of the underlying translation surface or rational billiard. The theme unifying the two components of the project is the technique of extracting useful information from the evolution of a discrete subset of Euclidean space under the action of a linear group.
The approach followed by the principal investigator is inspired by a dynamical systems viewpoint and has already led to breakthroughs in number theory. Further open problems are expected to be solved via this approach. The improved understanding of so-called semisimple flows that will result from the first part of the project can likely be used to develop efficient algorithms for generating rational approximations to irrational vectors and finding short vectors in lattices. These problems are of immense interest to computer scientists for their numerous applications, especially to cryptography. The investigation of nonergodic directions in the second part of the project is motivated in part by the recent discovery (by the principle investigator and his collaborators) of a striking phenomenon known as the "dichotomy of Hausdorff dimension" that has never before been observed in the dynamics of billiards (the term "billiards" here refers to a mathematical model for a certain type of collision, not to the activity one observes in pool halls). A better understanding of the mechanism that produces this phenomenon may potentially provide the basis for a new model to explain critical phenomena and may be of interest to physicists. On the human resource development side, the project involves the training of graduate students to become research mathematicians.
