Following a program initiated in 1979, the PI's project concerns some mathematical problems in the theory of aperiodic solids. The main tools used are coming from the Noncommutative Geometry, the theory of Random Matrices, the theory of Schrodinger operators, and the K-theory of C*-algebras. The project is divided into the following sections: (i) Phonons in Aperiodic Solids, (ii) Spectral Properties and Random Matrix Theory, (iii) Transverse Geometry of the Noncommutative Brillouin zone, (iv) Schrodinger's Operators on Tilings, (v) N-body Problem and Noncommutative Fermi Surface, and (vi) K-theory: toward a physical interpretation.
This program develops a self contained general theory liable to describe all possible solids from their microscopic structure. This framework is a unification of various point of views developed by physicists specializing in different types of materials. On the other hand, it brings into the theory the most advanced mathematical theories, such as Noncommutative Geometry, K-theory, Dynamical Systems, Noncommutative Probability, Random Matrices. It is a truly interdisciplinary program between Physics and Mathematics. Such an interdisciplinary program helps bring problems and ideas from one community to the other. K-theory is a tool that is appealing already in the field of string theory. This project shows that K theory applies also in other area of physics, where experiments are available. Conversely, the recent solution of the so-called "gap labelling conjecture" is an example of the impact of such research in mathematics.