The proposal is concerned with randomness in number theory .Randomness appears in many forms when studying diophantine problems and it is sometimes transparent ,with the burden being to establish the expected randomness, while in other cases discovering the randomness is already a major issue . The specific settings in which we seek to continue to investigate this phenomenon include diophantine problems connected with orbits of groups of integral affine morphisms ,the randomness in the such sequences as the Mobius function and especially its connection to low complexity dynamical systems ,the randomness in zeros of families of automorphic L-functions and in the zero sets (nodal lines) of the corresponding automorphic forms. Understanding the randomness issue is at the heart of many applications to number theory and arithmetic quantum chaos .
The interplay between number theory and other areas within mathematics has a long tradition.Many fields within mathematics were developed in part to solve problems concerning whole numbers and prime numbers. Naturally these areas continue to play a central role in the theory of numbers . More recent is the interplay between mathematical physics ,theoretical computer science and information theory .This cross fertilization is at the heart of this proposal .For example the modular surface gives perhaps the unique example of a classically chaotic hamiltonian for which the fine points of its quantization can be studied.Techniques from statistical physics and especially random matrix ensembles model precisely the zeros of families of central objects in number theory called "zeta functions" .Ideas from combinatorics and theoretical computer science ,notably "expander graphs", play a central role in understanding some classical number theoretical problems.
