MenuMathematical Physics and Harmonic Analysis Seminar
Spring 2021
| Date: | January 28, 2021 |
| Time: | 11:00am |
| Location: | Zoom |
| Speaker: | Ralph Kaufmann, Purdue University |
| Title: | Local Models and Global Constraints |
| Abstract: | Motivated by current research in condensed matter, we consider multi-parameter families of Hamiltonians which may have multiple Eigenvalues at so-called critical points. In this setting, we address several questions. In particular, for 3-d families with isolated critical points, we can give classification theorems for the local structures. We also show more generally that one can associate topological invariants to such systems, by various methods such as Chern classes and slicing. Finally, we investigate global constraints on the local structures and give theorems for the case in which the parameter space is a three-torus, which is relevant to real world periodic systems. We illustrate these constraints on a concrete system, the Gyroid, and its deformations. |
| Date: | January 29, 2021 |
| Time: | 2:00pm |
| Location: | Zoom |
| Speaker: | Chris Marx, Oberlin College |
| Title: | Potential dependence of the density of states: deterministic, ergodic, and random potentials |
| Abstract: | In this talk we will address the potential dependence of the density of states and related spectral functions for discrete Schr\"odinger operators on infinite graphs. Following ideas by J. Bourgain and A. Klein, we will consider the density of states {\em{outer}} measure (DOSoM), a {\em{deterministic}} quantity, which is well defined for {\em{all}} Schr\"odinger operators. We will explicitly quantify the potential dependence of the DOSoM in weak topology by proving a modulus of continuity with respect to the potential in $\ell{l}^\infinity$-norm. The resulting modulus of continuity reflects the geometry of the graph at infinity. For the special case of operators on $\mathbb{Z}^d$ our result implies Lipschitz continuity of the DOSoM, in the case of the Bethe lattice, we obtain that the DOSoM is $\frac{1}{2}$-log-H\"older continuous. Applications of this result to ergodic, and in further consequence, for random Schr\"odinger operators will be presented. This talk is based on joint work with Peter Hislop (University of Kentucky). |
| Date: | February 11, 2021 |
| Time: | 11:00am |
| Location: | Zoom |
| Speaker: | Marvin Plümer, FernUniversität in Hagen |
| Title: | Eigenvalue bounds for the Laplacian on embedded metric graphs |
| Abstract: | In this talk, we present some new developments in the theory of Laplacians on metric graphs. We discuss the role played by planarity or -- more generally -- the graph's genus and derive bounds for the eigenvalues of the Laplacian. For the derivation of these bounds, we make use of a recently developed transference principle by Amini and Cohen-Steiner that compares eigenvalues of continuous and discrete objects in a very convenient way. As a by-product of our methods we also obtain eigenvalue bounds for Laplacian on discrete weighted graphs. |
| Date: | February 12, 2021 |
| Time: | 2:00pm |
| Location: | Zoom |
| Speaker: | Benito Juarez-Aubry, IIMAS-UNAM |
| Title: | Semiclassical gravity in static spacetimes as a constrained initial value problem |
| Abstract: | Semiclassical gravity is the theory in which gravity is treated classically and matter is treated in the framework of quantum field theory. The spacetime metric tensor, which encodes gravitational effects in its curvature, interacts with matter through the semiclassical Einstein equations: matter sources the dynamics of the spacetime metric via the expectation value of the stress-energy tensor of the quantum fields, while the quantum fields propagate in curved spacetime. It is currently unknown whether semiclassical gravity has a well-posed initial value formulation even for free fields. In this talk, I will discuss the situation in static spacetimes, where time-translation symmetry greatly reduces the difficulty of the problem, and where one can show well-posedness. I will also discuss the main ideas on how to generalise these results to non-static spacetimes under some special circumstances. Based on arXiv:2011.05947 and on some unpublished work in progress with S. Modak. |
| Date: | February 25, 2021 |
| Time: | 11:00am |
| Location: | Zoom |
| Speaker: | Salma Lahbabi, ENSEM, UHII/MSDA, UM6P |
| Title: | A mean-field model for disordered crystals |
| Abstract: | In this talk, we consider disordered quantum crystals in the reduced Hartree- Fock (rHF) framework. The nuclei are supposed to be classical particles ar- ranged around a reference periodic configuration. In particular, we consider a family of nuclear distributions μ(ω, ·), where ω spans a probability space Ω. Under some assumptions on the nuclear distribution μ, the average energy per unit volume admits a minimizer, which is a solution of the rHF equations with short-range Yukawa interaction [2, 1]. We obtain partial results for Coulomb interacting systems. We also study localization properties of the mean-field Hamiltonian numerically [3]. References [1] Éric Cancès, Salma Lahbabi, and Mathieu Lewin. Mean-field electronic structure models for disordered materials. In Proceeding of the International Congress on Mathematical Physics, Aalborg (Denmark), August 2012. [2] Éric Cancès, Salma Lahbabi, and Mathieu Lewin. Mean-field models for disordered crystals. J. math. pures appl., 100(2):241274, 2013. [3] Salma Lahbabi. Étude mathématique de modèles quantiques et classiques pour les matériaux aléatoires à l'échelle atomique. PhD thesis, Université de Cergy-Pontoise, 2013. |
| Date: | March 4, 2021 |
| Time: | 11:00am |
| Location: | Zoom |
| Speaker: | Pavel Kurasov, Stockholm |
| Title: | Crystalline measures: from quantum graphs to stable polynomials |
| Abstract: | Quantum graphs surprised researchers by their extraordinary spectral properties and possibility to carry out explicit analysis. One such example is given by the trace formula connecting spectra of quantum graphs to the set of periodic orbits on the underlying metric graph (without any additional correction terms). It appears that the corresponding spectral measure provides an explicit example of so-called crystalline measures generalising classical Dirac comb and Poisson summation formula. Crystalline measures are tempered distributions given by locally finite purely atomic measures whose Fourier transform is also a purely atomic measure. Such measures were studied by J.-P. Kahane, A.-P. Guinand, and S. Mandelbrojt in the fifties. It remained unclear whether Dirac combs provide the only type of examples of crystalline measures with uniformly discrete support. We are going to show how to construct a wide family of crystalline measures using quantum graphs (differential operators on metric graphs) and more generally via stable polynomials. The measure we obtain are:
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| Date: | March 5, 2021 |
| Time: | 2:00pm |
| Location: | Zoom |
| Speaker: | Rostislav Grigorchuk, Texas A&M University |
| Title: | Spectra of groups and graphs: a short survey. |
| Abstract: | In my talk I will touch on such topics as the shape of the spectrum of Cayley and Schreier graphs of finitely generated groups, type of spectral measures, the question of A.Valette "Can one hear the shape of a group", and the relation to the random Schodinger operator. Based on numerous results with coauthors: L.Bartholdi, A.Zuk, Z.Sunic, D.Lenz, T.Nagnibeda, A.Perez, B.Simanek, A.Dudko and others. |
| Date: | March 11, 2021 |
| Time: | 11:00am |
| Location: | Zoom |
| Speaker: | Raphael Ducatez, UNIGE |
| Title: | Spectrum of critical Erdos Renyi graph |
| Abstract: | We analyze the spectrum of the (scaled) adjacency matrix A of the Erdős- Rényi graph G(N, d/N ) in the critical regime d = b log N. We establish a one-to-one correspondence between vertices of degree at least 2d and nontrivial eigenvalues outside the asymptotic bulk [−2, 2]. This correspondence implies a transition at an explicit b*. For d>b* log N the spectrum is just the bulk [−2, 2] and the eigenvectors are completely delocalized. For d< b* log N another phase appears. The spectrum outside [−2, 2] is not empty and there the eigenvectors concentrate around the large degree vertices.(joint work with Antti Knowles and Johannes Alt) |
| Date: | March 12, 2021 |
| Time: | 2:00pm |
| Location: | Zoom |
| Speaker: | Jon Harrison, Baylor University |
| Title: | Periodic orbit evaluation of a spectral statistic of quantum graphs without the semiclassical limit |
| Abstract: | Spectral statistics of classically chaotic quantum systems are often analyzed using their periodic orbit structure via Gutzwiller's trace formula, which holds in the semiclassical limit. We show that for chaotic 4-regular quantum graphs the variance of coefficients of the characteristic polynomial of the quantum evolution operator can be evaluated via periodic orbits without taking the semiclassical limit. The variance of the n-th coefficient is precisely determined by the number of primitive pseudo orbits (sets of distinct primitive periodic orbits) with n edges that fall in the following classes: those with no self-intersections, and those where all the self-intersections consist of two sections of the pseudo orbit crossing at a single vertex (2-encounters of length zero). |
| Date: | March 18, 2021 |
| Time: | 11:00am |
| Location: | Zoom |
| Speaker: | Silvius Klein, Puc-Rio |
| Title: | Mixed random-quasiperiodic Schrödinger operators |
| Abstract: | The purpose of this talk is to describe a discrete, one dimensional Schrödinger operator with mixed random-quasiperiodic potential and to discuss the positivity and the continuity of the corresponding Lyapunov exponent. This is part of a larger joint project with Ao Cai and Pedro Duarte (both from the University of Lisbon) in which we study the spectral properties of such operators as well as the stability under random perturbations of the Lyapunov exponent of quasiperiodic Schrödinger operators. |
| Date: | March 25, 2021 |
| Time: | 11:00am |
| Location: | Zoom |
| Speaker: | Dmitry Jakobson, Univ. Montreal |
| Title: | Nodal set and negative eigenvalues in conformal geometry |
| Abstract: | We first describe conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators (such as Yamabe or Paneitz operator). We discuss applications to curvature prescription problems. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension greater than 2. We show that 0 is generically not an eigenvalue of the conformal Laplacian. If time permits, we shall discuss related results on manifolds with boundary, as well as for weighted graphs. This is joint work with Y. Canzani, R. Gover, R. Ponge, A. Hassannezhad, M. Levitin, M. Karpukhin, G. Cox and Y. Sire. |
| Date: | March 26, 2021 |
| Time: | 2:00pm |
| Location: | Zoom |
| Speaker: | Sergei Tabachnikov, Penn State |
| Title: | Flavors of bicycle mathematics |
| Abstract: | This talk concerns a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich and has connections with several areas of research, including completely integrable systems. Here is a sampler of problems that I hope to touch upon: 1) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. 2) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are related by the bicycle (Backlund, Darboux) correspondence, which defines a discrete time dynamical system on the space of curves. This system is completely integrable and it is closely related with another, well studied, completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation. 3) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related with Ulam's problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions? This problem is also related to the motion of a charge in a magnetic field of a special kind. It turns out that the known solutions are solitons of the planar version of the filament equation. |
| Date: | April 2, 2021 |
| Time: | 2:00pm |
| Location: | Zoom |
| Speaker: | Peter Kuchment, Texas A&M University |
| Title: | The nodal sets mysteries |
| Abstract: | Nodal patterns of oscillating membranes have been known for hundreds of years. Leonardo da Vinci, Galileo Galilei, and Robert Hooke have observed them. By the nineteenth century they acquired the name of Chladni figures. Mathematically, they represent zero sets of eigenfunctions of the Laplace (or a more general) operator. In spite of such long history, many mysteries about these patterns (even in domains of Euclidean spaces, and even more on manifolds) still abound and attract recent attention of leading researchers working in physics, mathematics (including PDEs, math physics, and number theory) and even medical imaging. The talk will survey these issues, with concentration on the most recent results, where A&M mathematicians have been playing a significant role. No prior knowledge is assumed. |
| Date: | April 9, 2021 |
| Time: | 2:00pm |
| Location: | Zoom |
| Speaker: | Jared Wunsch, Northwestern University |
| Title: | Semiclassical analysis and the convergence of the finite element method |
| Abstract: | An important problem in numerical analysis is the solution of the Helmholtz equation in exterior domains, in variable media; this models the scattering of time-harmonic waves. The Finite Element Method (FEM) is a flexible and powerful tool for obtaining numerical solutions, but difficulties are known to arise in obtaining convergence estimates for FEM that are uniform as the frequency of waves tends to infinity. I will describe some recent joint work with David Lafontaine and Euan Spence that yields new convergence results for the FEM which are uniform in the frequency parameter. The essential new tools come from semiclassical microlocal analysis. No knowledge of either FEM or semiclassical analysis will be assumed in the talk, however. |
| Date: | April 16, 2021 |
| Time: | 2:00pm |
| Location: | Zoom |
| Speaker: | Cosmas Kravaris, Texas A&M University |
| Title: | On the density of eigenvalues on discrete periodic graphs |
| Abstract: | Using the Floquet-Bloch transform, we show that Zd-periodic graphs have finitely many finite support eigenfunctions up to translations and linear combinations and show that this can be used to calculate the density of eigenvalues. We study the Kagome lattice to illustrate these techniques and generalize the claims to amenable quasi-homogeneous graphs whose acting group has Noetherian group algebra (this includes all virtually polycyclic groups). Finally, we provide a formula for the von Neumann dimension (i.e. density) of eigenvalues on Zd-periodic graphs using syzygy modules. |
| Date: | April 22, 2021 |
| Time: | 11:00am |
| Location: | Zoom |
| Speaker: | Mikael Sundqvist, Lund University |
| Title: | Spectral flow for pair compatible equipartitions |
| Abstract: | In this talk we discuss how a recent spectral flow approach, proposed by Berkolaiko–Cox–Marzuola for analyzing the nodal deficiency of the nodal partition associated to an eigenfunction, can be extended to more general partitions. To be more precise, we discuss how the method can be extended to spectral equipartitions that satisfy a pair compatible condition. Nodal partitions and spectral minimal partitions are examples of such partitions. Based on joint work with Bernard Helffer. |
| Date: | April 23, 2021 |
| Time: | 2:00pm |
| Location: | Zoom |
| Speaker: | Jason Metcalfe, University of North Carolina at Chapel Hill |
| Title: | Local energy in the presence of degenerate trapping |
| Abstract: | Trapping is a known obstruction to local energy estimates for the wave equation and local smoothing estimates for the Schrödinger equation. When this trapping is sufficiently unstable, it is known that estimates with a logarithmic loss can be obtained. On the other hand, for very stable trapping, it is known that all but a logarithmic amount of local energy decay is lost. Until somewhat recently, explicit examples of scenarios where an algebraic loss (of regularity) was both necessary and sufficient for local energy decay had not be constructed. We will review what is known in these specific examples. We will also examine the relationship between the trapping and the existence of a boundary. In this highly symmetric case, a relatively simple proof showing a bifurcation in the behavior of local energy as the boundary passes through the trapping is available. This is related, e.g., to the instability of ultracompact neutrino stars. |
| Date: | April 29, 2021 |
| Time: | 11:00am |
| Location: | Zoom |
| Speaker: | Constanza Rojas-Molina, CY Cergy Paris Université |
| Title: | Fractional random Schrödinger operators, integrated density of states and localization |
| Abstract: | In this talk we will review some recent results on the fractional Anderson model, a random Schrödinger operator driven by a fractional laplacian. The interest on the latter lies in their association to stable Levy processes, random walks with long jumps and anomalous diffusion. We discuss in this talk the interplay between the non-locality of the fractional laplacian and the localization properties of the random potential in the fractional Anderson model, in both the continuous and discrete settings. In the discrete setting we study the integrated density of states and show a fractional version of Lifshitz tails. This coincides with results obtained in the continuous setting by the probability community. This is based on joint work with M. Gebert (LMU Munich). |
