We consider 2d random ergodic magnetic Schrödinger operators on domains with and without boundary. By extending the gauge covariant magnetic perturbation theory to infinite domains with boundary, we prove that the celebrated bulk-edge correspondence of systems with a (mobility) gap at zero temperature, i.e. the equality of the transversal bulk conductivity and the edge conductance, holds also at all positive temperatures and irrespective of a (mobility) gap in the bulk. While the quantization of the transverse bulk conductivity and the edge conductivity at zero temperature and in the presence of a (mobility) gap is a topological feature, their equality is not, but applies much more generally. Moreover, we obtain a formula which states that at any positive temperature, the derivative of a large class of bulk partition functions with respect to the external constant magnetic field is equal to the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge. Physically, our formula implies not only equality between transverse bulk conductivity and edge conductance, but also equality between bulk magnetization density and edge current at all temperatures. As a corollary of our purely analytical arguments we find that in gapped systems the transverse bulk conductivity and the edge conductance approach their quantized (integer) values with a rate that is exponential in the inverse temperature.

This lecture summarizes what the Theory of Functional Connections (TFC) is and presents the most important applications to date. The TFC performs linear functional interpolation. This allows to derive analytical expressions with embedded constraints, expressions describing all possible functions satisfying a set of constraints. These expressions are derived for a wide class of constraints, including points and derivatives constraints, relative constraints, linear combination of constraints, component constraints, and integral constraints. An immediate impact of TFC is on constrained optimization problems as the whole search space is reduced to just the space of solutions fully satisfying the constraints. This way a large set of constrained optimization problems can be transformed in unconstrained problems, allowing more simple, fast, reliable, and accurate methods to solve them. For instance, TFC allows to obtain fast and machine-error accurate solutions of linear and nonlinear ordinary differential equations. TFC has been extended to n-dimensions (Multivariate TFC). This allows to derive numerical methods to solve partial and stochastic differential equations. This lecture also provides some other TFC applications as, for instance, to homotopy continuation, calculus of variation, nonlinear programming, and optimal control (energy-efficient optimal landing on large bodies). Location: Meeting id: 980 8118 3032 Passcode: mpf21 Join Zoom Meeting https://tamu.zoom.us/j/98081183032?pwd=WitENWJqWjRyWVQvU3RQZDd4Mm9sUT09
Abstract

Floquet topological insulators are an emerging category of materials whose properties are transformed by time-periodic forcing. Can their properties be understood from their first-principles continuum models, i.e., from a driven Schrodinger equation? First, we study the transformation of graphene from a conductor into an insulator under a time-periodic magnetic potential. We show that the dynamics of certain wave-packets are governed by a Dirac equation, which has a spectral gap property. This gap is then carried back to the original Schrodinger equation in the form of an “effective gap” - a new and physically-relevant relaxation of a spectral gap. Next, we consider periodic media with a localized defect, and ask whether edge/defect modes remain stable under forcing. In a model of planar waveguides, we see how such modes decay and disappear due to resonant coupling with the radiation modes.

I will describe some problems relating to machine learning and their connections to random matrix theory and spin glasses. These connections give a mathematical framework for understanding in qualitative terms the effectiveness of certain algorithms that are important in machine learning, but developing them into precise models remains a major challenge. I will reflect on the different roles played by models in computer science and physics, focusing on those involving random matrices.

In this talk I present a rigorous derivation of Bogoliubov theory for systems of $N$ trapped bosons in $\mathbb{R}^3$ in the so called Gross-Pitaevskii regime, characterized by a scattering length of order $N^{-1}$. We prove complete Bose-Einstein condensation for approximate ground states with optimal rate and determine the low-energy excitation spectrum of the system up to errors vanishing in the limit $N\to \infty$. The talk is based on joint work with S. Schraven and B. Schlein.

In the 80s, the physicists Rammal and Tolouse observed that suitable series of eigenvalues in the finite graph approximations of the Sierpinski gasket produced an orbit of a particular dynamical system. That observation lead to a complete description of the spectrum of the standard Laplace operator by Fukushima and Shima. The study of this spectrum has since then revealed structures with many interesting features not seen in other more classical settings. For instance, it presents large exponential gaps (or spacings), whose existence and properties have extensively been studied. What happens with the small gaps? This fairly challenging question had eluded previous investigations and is the main subject of the present talk, where we discuss yet another remarkable fact: Any two consequent eigenvalues in the Dirichlet or in the Neumann spectrum of the Laplacian on the Sierpinski gasket are separated at least by the spectral gap.