This proposal comprises three major projects and several additional smaller projects. The first of the three main projects concerns the approximation of real numbers by rationals, a classical subject of research in number theory. Over the past few decades many authors have applied ad hoc methods to obtain highly nontrivial bounds for the approximation by rationals whose denominators belong to particular types of sequences, such as the sequence of prime numbers, or the sequence of squares. The proposer believes that results of this type should hold in a much more general context, and he hopes to prove a universal result for general sequences that is of nearly the same quality as the results that have been obtained in the literature for special sequences. The second project concerns the distribution of zeros of L-functions, a vast area of research motivated by the Riemann Hypothesis and its generalizations. In particular, the so-called pair-correlation for the sequence of such zeros has received much attention in recent years because of surprising connections with phenomena arising in quantum chaos and the theory of random matrices. The proposer plans to investigate a new type of pair correlation, in which a second sequence is introduced, in the hope of obtaining new results by exploiting the added flexibility provided by the second sequence. The third project is to apply methods and tools from number theory to certain problems in mathematical physics, in particular, the so-called Lorentz gas and the Sinai billiard problem.
This project falls in the area of number theory, one of the oldest subjects of mathematics which in recent years has received renewed interest due to newly found applications (e.g., to cryptography and coding theory), unexpected connections with physics (such as the theory of quantum chaos and statistical mechanics), and major unsolved problems (such as the Riemann Hypothesis, one of the seven "million dollar" problems in mathematics). The proposed research is concerned with several problems that fall at the interface of number theory and mathematical physics and build upon ongoing work by the proposer and his collaborators. It is hoped that deploying the powerful methods and tools of number theory to these problems will lead toinsights and results that specialists have not been able to obtainand which will be of interest to a broad range of researchers, both in mathematics and in physics.
