Excellence in Research: Numerical Analysis of Quasiperiodic Topology

Howard University, Washington DC

Investigators

Abstract

Hamiltonian dynamical systems are of paramount importance in both Mathematics and Physics. These systems are defined by an energy function, called Hamiltonian, and often admit first integrals, namely other functions, independent on the Hamiltonian but that also, just like the Hamiltonian, do not change their value over the solutions of the equations of motion. Since this formalism was introduced by Sir W.R. Hamilton almost two centuries ago, it grew to embrace the physics of most non-dissipative phenomena, as well as several important branches of geometry that arose naturally out of it, and it became one of the main building block of quantum mechanics. Nevertheless, until relatively recently, an important class of Hamiltonian systems was overlooked, namely the case when some first integral is multivalued (an example of multivalued function is the angle coordinate on a circle). Such systems arise naturally from quantum mechanics (in the so-called semiclassical approximation), in particular in the theory of conductivity of metals at low temperature under a strong magnetic field. Experimental data on this phenomenon were collected for many metals starting from seventy years ago but they could not be checked against the theory exactly for the lack of a theory of Hamiltonian systems with multivalued Hamiltonians. Since the Eighties, Fields medalist S.P. Novikov and his school started filling this gap and about ten years ago the experimental data were successfully verified in the simplest cases, namely for Au and Ag. At the same time, a new field of topology, called Quasiperiodic topology, arose as the generalization and formalization of this Hamiltonian system and important theoretical results have been found after the numerical study of the simplest cases. This project aims, on the one hand, at completing the verification of these fundamental experimental data for the several metals still unchecked and, on the other hand, at extending and deepening the numerical study of quasiperiodic topology. The main goal of this project is to continue and deepen the numerical study of Quasiperiodic Topology focusing, in particular, on the topology of level sets of multivalued maps on the n-dimensional tori. Its main subgoal is the contextual analytical study of the properties suggested by the numerical experiments. This will be achieved through the following specific objectives and methods: (1) development and implementation in Perl/Python/C++ of old and new algorithms for the numerical and analytical study of multivalued maps on n-tori; (2) numerical exploration and classification of these maps; (3) analytical study of minor and major properties of such maps. As an application of these results, a further important subgoal of the project is the application of the code generated in (1) to verify from first principles for the first time some important experimental data about conductivity in metals measured in Fifties and Sixties and whose mathematical description involves quasiperiodic functions. 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.