Diophantine Approximation and Aperiodic Order

University Of Houston, Houston TX

Investigators

Abstract

This award aims to develop and harvest a collection of interrelated results spanning the fields of number theory, dynamical systems, and aperiodic order. A central theme is to explore problems which have known applications to the mathematics underpinning models for physical materials called quasicrystals. The research goals are to use tools from number theory, probability, topological dynamics, and ergodic theory, to prove new results about a number of long standing open problems with significance in the world of pure mathematics, which also have the potential for real world applications. The Principal Investigarot will mentor graduate students on topics related to this award. Quasicrystals are physical materials with highly ordered molecular structures, causing them to produce pure point diffraction patterns, but which also possess rotational symmetries that are forbidden by the classical Crystallographic Restriction Theorem. The discovery of these materials in the 1980's by Dan Shechtman revolutionized the world of crystallography and later earned Shechtman a Nobel prize. In the mathematical world the study, and even the existence, of quasicrystals is intimately related to the theory of aperiodic tilings of Euclidean space. Quasicrystals are often modeled by cut and project sets, which give an abundance of examples of such tilings and which, generically, are good examples of systems which possess `aperiodic order'. In the last decade there has been an explosion of insight linking problems in number theory with problems about cut and project sets. A number of unsolved problems in Diophantine approximation have been reformulated, and some have been solved, using ideas from the theory of mathematical quasicrystals. Major problems about sets with aperiodic order have also been solved by understanding the reverse connections. The research in this project is centered around well-known open problems such as the Littlewood Conjecture and the Pisot Conjecture, which are sitting on the border of these subjects. It aims to develop the theory around them in order to create progress in both the realms of pure and applied mathematics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.