RII Track-4: Sharp arithmetic transitions and universality in one-frequency quasiperiodic systems

University Of Mississippi, University MS

Investigators

Abstract

Non-technical description Mathematics provides essential tools for the study and description of many natural phenomena. An important area of research in mathematics is the development of tools to study the movement of different systems (termed dynamical phenomena), which range from the movement of celestial bodies to heart function and fluctuations in the stock market. This fellowship will initiate a new collaboration between the University of Mississippi (UM) and University of California Irvine (UCI), enabling an extended research visit at UCI. The project will develop state of the art tools for study of dynamical systems, which will allow scientists to probe systems at different spatial and time scales, revealing properties of systems that are universal. An example of the type of research that has relied on these tools in the past is the description of the transition between the liquid and gas phases (i.e., boiling and evaporation). A particular focus of this project will be on sharp transitions and universality in two types of systems: relatively simple systems that underlie more complicated systems; and systems arising from quantum physics. This project will lead to advancement of both areas, strengthen the research program in dynamical systems and mathematical physics at UM, and enhance its undergraduate and graduate education. Technical description This project involves cutting-edge problems at the interface of dynamical systems and mathematical physics, in particular the spectral theory of Schrödinger operators, which will be approached using the powerful idea of renormalization. The project will focus on sharp arithmetic transitions and universality in quasiperiodic systems, specifically circle maps with breaks and Schrödinger operators. Circle maps with an irrational rotation number are a prototype of quasiperiodic dynamics and are closely related to many other systems, including the Schrödinger cocycles arising from the spectral analysis of Schrödinger operators. The objectives of the project are two-fold: (i) to seek a sharp arithmetic transition for the rigidity of circle maps with a break, i.e., the optimal set of rotation numbers for which a smooth conjugacy between any two maps with the same rotation number in that set, and the same size of the break, is guaranteed; and (ii) to develop a renormalization approach towards explaining the self-similarity of the Hofstadter butterfly and universality in the behavior of eigensolutions, arising when studying the spectrum of one-frequency quasiperiodic Schrödinger operators. Both problems will significantly advance the knowledge in these fields. The second problem would represent significant advancement not only in mathematics, but also in related areas of physics.