MenuMathematical Physics and Harmonic Analysis Seminar
Fall 2021
| Date: | September 8, 2021 |
| Time: | 10:00am |
| Location: | Zoom |
| Speaker: | Horia Cornean, Aalborg University |
| Title: | Generalized bulk-edge correspondence at positive temperature |
| Abstract: | 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. |
| Date: | September 10, 2021 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Agam Shayit, TAMU |
| Title: | Vacuum energy density and pressure inside a soft wall |
| Abstract: | In the study of quantum vacuum energy and the Casimir effect, it is desirable to model the conductor by a potential of the form V(z) = z^α. Unlike the standard Dirichlet wall, this model does not violate the principle of virtual work under regularization. Previously, this ``soft wall" model was formalized for a massless scalar field, and the expectation value of the stress tensor was expressed in terms of the reduced Green function of the equation of motion. In the limit of interest α >> 1, which corresponds to the Dirichlet wall, a closed-form expression for the reduced Green function cannot be found. Here we develop a piecewise approximation scheme incorporating the perturbative and WKB expansions of the Green function, as well as an interpolating spline in the region where neither expansion is valid. We then apply the scheme to the sextic soft wall and use it to compute the renormalized energy density and pressure inside the cavity for various conformal parameters. The consistency of the results is verified by comparison to their numerical counterparts and verification of the trace anomaly and the conservation law. Finally, we use the approximation scheme to reproduce the energy density inside the quadratic wall, which was previously calculated exactly. |
| Date: | September 17, 2021 |
| Time: | 1:50pm |
| Location: | Zoom |
| Speaker: | Daniele Mortari, TAMU |
| Title: | The Theory of Functional Connections |
| Abstract: | 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 |
| Date: | October 6, 2021 |
| Time: | 10:00am |
| Location: | Zoom |
| Speaker: | Ivan Veselic, Dortmund (Germany) |
| Title: | Scale free unique continuation estimates and applications for periodic and random operators |
| Abstract: | With Ivica Nakic, Matthias Taeufer and Martin Tautenhahn we established a quantitative unique continuation estimate for spectral projectors of Schroedinger operators. It compares the L^2 norm of a function in a spectral subspace associated to a bounded energy interval to the L^2 norm on an equidistributed set. These estimates allow to give quantitative two-sided bounds on the lifting of edges of bands of essential spectrum, as well as on discrete eigenvalues between two such bands. It also allows to deduce Anderson localization in regimes where this was not possible before. For instance, Albrecht Seelmann and Matthias Taeufer showed that Anderson localization occurs at random perturbations of band edges of periodic potentials, whether the edges exhibit regular Floquet eigenvalue minima or not. |
| Date: | October 15, 2021 |
| Time: | 1:50pm |
| Location: | BLO306&Zoom |
| Speaker: | Amir Sagiv, Columbia University |
| Title: | Floquet Hamiltonians - effective gaps and resonant decay |
| 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. |
| Date: | October 20, 2021 |
| Time: | 10:00am |
| Location: | Zoom |
| Speaker: | Noema Nicolussi, Ecole Polytechnique |
| Title: | Asymptotics of Green functions: Riemann surfaces and Graphs |
| Abstract: | There are many interesting parallels between the analysis and geometry of Riemann surfaces and graphs. Both settings admit a canonical measure/metric (the Arakelov--Bergman and Zhang measures) and the associated canonical Green function reflects crucial geometric information. Motivated by the question of describing the limit of the Green function on degenerating Riemann surfaces, we introduce new and higher rank versions of metric graphs and their Laplace operators. We discuss how these limit objects describe the asymptotic of solutions to the Poisson equation and, in particular, the Green function on metric graphs and Riemann surfaces close to the boundary of their respective moduli spaces. Based on joint work with Omid Amini (Ecole Polytechnique). |
| Date: | October 27, 2021 |
| Time: | 10:00am |
| Location: | Zoom |
| Speaker: | Jon Keating, University of Oxford |
| Title: | Random matrices, spin glasses, and machine learning |
| Abstract: | 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. |
| Date: | October 29, 2021 |
| Time: | 1:50pm |
| Location: | Zoom |
| Speaker: | Burak Hatinoglu, UC Santa Cruz |
| Title: | Spectral properties of the periodic 4th order Schrodinger operator on the hexagonal lattice |
| Abstract: | This talk will focus on the 4th order Schr\"{o}dinger operator $d^4/dx^4 + q(x)$ on the hexagonal lattice (graphene) and its geometric perturbations with self-adjoint vertex conditions and a real periodic symmetric potential $q$. I will consider the following spectral properties of this Hamiltonian on graphene: absolutely continuous, singular continuous and pure-point spectra, dispersion relation, singular Dirac points and energy levels of reducible Fermi surfaces. I will also discuss some of these spectral properties for the same operator on lattices in the geometric neighborhood of graphene. This talk is based on a joint work with Mahmood Ettehad (University of Minnesota). |
| Date: | November 10, 2021 |
| Time: | 10:00am |
| Location: | Zoom |
| Speaker: | Christian Brennecke, University of Bonn |
| Title: | Bogoliubov Theory for Trapped Bosons in the Gross-Pitaevskii Regime |
| Abstract: | 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. |
| Date: | November 19, 2021 |
| Time: | 1:50pm |
| Location: | Zoom |
| Speaker: | Laura Shou, Princeton University |
| Title: | Pointwise Weyl law for graphs from quantized interval maps |
| Abstract: | In this talk I will discuss the eigenvectors of families of unitary matrices obtained from quantization of one-dimensional interval maps. This quantization for interval maps was introduced by Pakoński et al. [J. Phys. A: Math. Gen. 34 9303 (2001)] as a model for quantum chaos on graphs. The resulting unitary matrices are sparse, yet numerically exhibit CUE random matrix behavior. To analyze the eigenvectors, I prove a pointwise Weyl law with shrinking spectral windows. This implies a stronger version of the quantum ergodic theorem for these models, and shows in the semiclassical limit that a family of randomly perturbed quantizations has approximately Gaussian eigenvectors. |
| Date: | December 3, 2021 |
| Time: | 1:50pm |
| Location: | Zoom |
| Speaker: | Patricia Alonso Ruiz, TAMU |
| Title: | Minimal eigenvalue spacing in the Sierpinski gasket |
| Abstract: | 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. |
| Date: | December 8, 2021 |
| Time: | 10:00am |
| Location: | BLOC 628 |
| Speaker: | Jacob Shapiro, Princeton |
| Title: | Delocalization in the integer-valued Gaussian Field and the BKT phase of the 2D Villain model |
| Abstract: | It is shown that the Villain model of two-component spins over two dimensional lattices exhibits slow, non-summable, decay of correlations at any temperature at which the dual integer-valued Gaussian field exhibits delocalization. For the latter, we extend the recent proof by Lammers of a delocalization transition in two dimensional graphs of degree three, to all doubly periodic graphs, in particular to Z^2. Taken together these two statements yield a new perspective on the BKT phase transition in the Villain model, and a new proof on delocalization in two dimensional integer-valued height functions. Joint with Aizenman, Harel and Peled. |
| Date: | December 10, 2021 |
| Time: | 1:50pm |
| Location: | Zoom |
| Speaker: | Spiridoula Matsika, Temple University |
| Title: | Conical intersections in quantum chemistry |
| Abstract: | In the quantum mechanical treatment of molecules we use the Born-Oppenheimer (adiabatic) approximation, in which the motion of nuclei and electrons is separated because of their large difference in masses. In this approximation the coupling between different electronic potential energy surfaces (PES) is neglected and nuclei move on a single electronic PES. Modeling the motion of nuclei on PESs allows us to model the structure of molecules, their spectroscopy, and chemical reactions. Nevertheless, non-adiabatic processes where the coupling between different PES becomes large are important and ubiquitous in photochemical and other reactions. These processes are facilitated by the close proximity of PESs, and especially by the extreme case where the PESs become degenerate forming conical intersections. In this talk we will discuss the description and basic applications of conical intersections in chemical problems. |
