The first project, quantum disordered systems and random matrices, aims to explore the connections between the random matrix and random Schrodinger equations. The fundamental questions in these subjects are the Wigner-Dyson universality for the eigenvalue gap distribution and the extended state conjecture for the random Schrodinger equations. Since the extended state conjecture is out of the reach of the current technique, we propose to estimate the mean square displacement for the random Schrodinger equations in long time scaling limits in two cases: 1. The random potential is time dependent and has a short time memory. 2. The random potential is given by a phonon field in equilibrium. Inspired by the same extended state conjecture, we propose to study the localization/delocalization property of eigenvectors of random matrices. Finally, as the first step toward the Wigner-Dyson universality, we propose to estimate the difference between the density of eigenvalues of random matrices in microscopic windows and the Wigner semicircle law. The second project, dynamics of Bose gas, concerns the derivation of the the phenomenological Gross-Pitaevskii equation from the many-body Schrodinger equations. We also propose to estimate precisely the errors between the Gross-Pitaevskii equation and the many-body Schrodinger equation. The third project, regularity of axisymmetric incompressible Navier-Stokes equations, aims to prove the regularity of the INS for the axisymmetric flow under certain mild assumptions.
This first project of this proposal is aimed to establish the conducting properties of semiconductors and other disordered systems. The mathematical model for these systems in the simplest form is given by matrices with random entries. Our first project is designed to provide rigorous proof that conduction does occur in the random matrix models. The second project aim to establish the Gross-Pitaevskii equation---the fundamental equation governing the dynamics of the Bose-Einstein condensate (a recent discovered material but was theorized by Bose and Einstein almost a century ago). This project aims to lay rigorous foundation for the description of the dynamics of these systems. The third project concerns regularity of the incompressible Navier-Stokes equations for axisymmetric flow. This project will not only lead to deeper understanding of the INS equations in general, but the axisymmetric flow is important for modeling atmospheric events such as hurricanes and tornados. Most problems covered in this proposal were studied in their own traditionally fields such as random matrix, mathematical physics, condense matter physics or partial differential equations. This proposal aims to establish connections between these subjects and to train students and postdoctors with interdiscipline technique and broad scientific perspectives. It will also lead to collaborations of researchers in these diverse disciplines.
Many-Body Quantum Dynamics and Quantum Disorder Systems, 0804279, H.-T. Yau Intellectual Merit Wigner’s grand vision that the local spectral statistics of large correlated quantum systems are modeled by random matrix statistics is a groundbreaking idea in the history of probability theory and statistical physics. Classical probability theory was built upon modeling systems with no or weak correlations; however, random matrix statistics are the only known general laws for highly correlated systems. These laws appear in large classes of random matrix ensembles, in the spectral statistics of random Schr¨odinger operators, zeros of Riemann zeta functions, and in many other examples. The fundamental mystery is why the random matrix laws are ubiquitous. A special case of this grand vision is the Wigner-Dyson universality conjecture, which states that the local spectral statistics of random matrices are independent of the matrix laws. The main outcome of this project was successfully solving this universality conjecture for both Wigner ensembles and β ensembles. The main intellectual merit was the establishment of the link between the universality conjecture and Dyson’s conjecture, which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order N −1 with N the size of the matrices. We proved Dyson’s conjecture to be correct and from this result we solved the Wigner-Dyson conjecture. Our analysis of Dyson Brownian motion was derived from the classical ideas of De Giorgi-Nash-Moser on the regularity of parabolic equations. Broader Impacts This project brought together experts from the fields of probability theory, statistics, combi- natorics, mathematical physics, and partial differential equations. It fostered close collaborations between researchers from these diverse disciplines. During the project, the PI trained students and post-doctoral fellows to utilize interdisciplinary techniques and provided them with broad scientific perspectives. There were three postdoctors, A. Knowles, P. Bourgade and K. Schnelli, trained in this project. Two Ph-D students, J. Lee and Benjamin Stetler, were trained in this project. There were also more than five undergraduate theses produced in this project.
