# Mathematical Physics and Harmonic Analysis Seminar

Date Time |
Location | Speaker | Title – click for abstract | |
---|---|---|---|---|

09/0810:00am |
Zoom | Horia Cornean Aalborg University |
Generalized bulk-edge correspondence at positive temperatureWe 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. | |

09/101:50pm |
BLOC 628 | Agam Shayit TAMU |
Vacuum energy density and pressure inside a soft wallIn 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. | |

09/171:50pm |
Zoom | Daniele Mortari TAMU |
The Theory of Functional ConnectionsThis 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 | |

10/0610:00am |
Zoom | Ivan Veselic Dortmund (Germany) |
Scale free unique continuation estimates and applications for periodic and random operatorsWith 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. | |

10/151:50pm |
BLO306&Zoom | Amir Sagiv Columbia University |
Floquet Hamiltonians - effective gaps and resonant decayFloquet 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.
| |

10/2010:00am |
Zoom | Noema Nicolussi Ecole Polytechnique |
Asymptotics of Green functions: Riemann surfaces and GraphsThere 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). | |

10/2710:00am |
Zoom | Jon Keating University of Oxford |
Random matrices, spin glasses, and machine learningI 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. | |

10/291:50pm |
Zoom | Burak Hatinoglu UC Santa Cruz |
TBA | |

11/1010:00am |
Zoom | Christian Brennecke University of Bonn |
Bogoliubov Theory for Trapped Bosons in the Gross-Pitaevskii RegimeIn 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. | |

11/191:50pm |
Zoom | Laura Shou Princeton University |
TBA | |

12/031:50pm |
TBD | Patricia Alonso Ruiz TAMU |
Minimal eigenvalue spacing in the Sierpinski gasketIn 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. |

The organizers for this seminar are Bob Booth and Rodrigo Matos. Email them with talk suggestions and to request the Zoom link.