MenuMathematical Physics and Harmonic Analysis Seminar
Fall 2019
| Date: | August 30, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | P. Kuchment |
| Title: | CANCELLED |
| Abstract: | due to the funeral of Professor Michael Boshernitzan (Rice U.) Analyticity of the spectrum and Dirichlet-to-Neumann operator technique for quantum graphs. We establish the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters over the whole Grassmannian of possible vertex conditions. We also discuss the Dirichlet-to-Neumann (DtN) technique of relating quantum and discrete graph operators, which allows one to transfer some results from the discrete to the quantum graph case, but which has issues at the Dirichlet spectrum. We conclude that this difficulty stems from the use of specific coordinates in a Grassmannian and can be easily avoided. Joint work with Jia Zhao (Hebei University of Technology, China) |
| Date: | September 6, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Robert Booth, Texas A&M University |
| Title: | Almost Global Existence for Asymptotically Euclidean Quasilinear Wave Equations |
| Abstract: | In this talk, we will discuss a recent result demonstrating almost global existence for a class of non-trapping asymptotically Euclidean quasilinear wave equations with small initial data. A novel feature is that the wave operator may be a large perturbation of the usual D'Alembertian operator. The solution is constructed via an iteration argument based on local energy estimates for an appropriately linearized version of our wave equation. Techniques used to develop the key local energy estimate include microlocal analysis, Carleman estimates, and positive commutator arguments. |
| Date: | September 12, 2019 |
| Time: | 2:50pm |
| Location: | BLOC 220 |
| Speaker: | Burak Hatinoglu, TAMU |
| Title: | A complex analytic approach to mixed spectral problems (Unusual day, time and room) |
| Abstract: | We consider a generalization of classical inverse spectral results of Borg and Marchenko for the Schroedinger operators on a finite interval with an L^1-potential. After a brief review of inverse spectral theory of one dimensional regular Schroedinger operators, we will discuss the following problem: Can one spectrum together with subsets of another spectrum and norming constants recover the potential? |
| Date: | September 13, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Selim Sukhtaiev, Rice University |
| Title: | Hadamard-type formulas and resolvent expansions |
| Abstract: | In this talk I will discuss an Hadamard-type variation formula for eigenvalue curves of one-parameter families of self-adjoint operators. The main application will be given to computation of the spectral flow of differential operators in terms of the signature of the Maslov form. This is joint work (in progress) with Y. Latushkin. |
| Date: | September 20, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Junho Yang, Texas A&M University, Statistics |
| Title: | Rate of convergence in Szego limit theorems |
| Abstract: | We discuss an improved rate of convergence of eigenvalues of Hermitian Toeplitz matrices originated from Szego's limiting eigenvalue distribution theorem. The proof consists of two steps: 1) provide point-wise asymptotics of all eigenvalues of banded Hermitian matrices; 2) Approximate general n-by-n Toeplitz matrix with p-diagonal banded Hermitian matrix in the sense of the "best predictive" function, and provide an entrywise 1-norm error bound. Our results can be used to calculate the exact bound of the Whittle's likelihood approximation, and approximations for the inverse of Toeplitz matrices. Joint work with Suhasini Subbarao (TAMU Statistics). |
| Date: | September 25, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Lior Alon, Technion - Israel Institute of Technology |
| Title: | On a CLT conjecture for the nodal statistics of quantum graphs |
| Abstract: | Understanding statistical properties of Laplacian eigenfunctions in general and their nodal sets in particular have an important role in the field of spectral geometry, and interest both mathematicians and physicists. A quantum graph is a system of a metric graph with self adjoint Schrodinger operator acting on it. In the case of quantum graphs it was proven that the number of points on which each eigenfunction vanish also known as the nodal count is bounded away from the spectral position of the eigenvalue by a topological quantity, the first Betti number of the graph. A remarkable result by Berkolaiko and Weyand (with another proof for discrete graphs by Colin de Verdiere) showed that the nodal surplus is equal to a magnetic stability index of the corresponding eigenvalue. Both from the nodal count point of view and from the physical magnetic point of view, it is interesting to consider the distribution of these indices over the spectrum. In our work we show that such a density exist and define a nodal surplus distribution. Moreover this distribution is symmetric, which allows to deduce the Betti number of a graph from its nodal count. A further result proves that the distribution is binomial with parameter half for a certain large family of graphs. The binomial distribution satisfy a CLT convergence, and a numerical study indicates that the CLT convergence is independent of the specific choice of the growing family of graphs. In my talk I will talk about our latest results extending the number of families of graphs for which we can prove the CLT convergence. Joint work with Ram Band and Gregory Berkolaiko. |
| Date: | September 27, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Blake Keeler, UNC |
| Title: | Random Waves and the Spectral Function on Manifolds without Conjugate Points |
| Abstract: | In this talk, we discuss off-diagonal Weyl asymptotics on a compact manifold M, with the goal of understanding the statistical properties of monochromatic random waves. These waves can be thought of as randomized "approximate eigenfunctions," and their statistics are completely determined by an associated covariance kernel which coincides exactly with a rescaled version of the spectral function of the Laplace-Beltrami operator. We will prove that in the geometric setting of manifolds without conjugate points, one can obtain a logarithmic improvement in the two-point Weyl law for this spectral function, provided one restricts to a shrinking neighborhood of the diagonal in M x M. This then implies that the covariance kernel of a monochromatic random wave locally converges to a universal limit at a logarithmic rate as we take the frequency parameter to infinity. This result generalizes the work of Berard, who obtained the logarithmic improvement in the on-diagonal case for manifolds with nonpositive curvature. |
| Date: | October 4, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Wencai Liu, Texas A&M University |
| Title: | Anderson localization for multi-frequency quasi-periodic operators on higher dimensional latices |
| Abstract: | The first part of the talk, based on a joint work with S. Jitomirskaya and Y. Shi, is devoted to study multi-frequency quasi-periodic operators on higher dimensional lattices. We establish the Anderson localization for general analytic $k$-frequency quasi-periodic operators on $\Z^d$ for arbitrary $k, d$. This is a generalization of Bourgain-Goldstein-Schlag's result $b=d=2$ and Bourgain's result $b=d\geq 3$. Our proof works for Toeplitz operators as well. In the second part of the talk, I will discuss several closely related topics. For example, 1. Use the quantitative unique continuation to establish the Anderson localization of random Schr\"odinger operators with singular distributions. 2. Use rotation $C^{\star}$ algebra to tackle the dry ten Martini problem (gap labelling theorem). 3. Use the machinery of proof of Anderson localization to construct KAM (Kolmogorov-Arnold-Moser) tori for NLS and NLW equations. |
| Date: | October 9, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 624 |
| Speaker: | Michael Levitin, University of Reading (UK) |
| Title: | Asymptotics of Steklov eigenvalues for curvilinear polygons (Unusual date and room!) |
| Abstract: | I will discuss sharp asymptotics of large Steklov eigenvalues for curvilinear polygons. The asymptotic expressions for eigenvalues are given in terms of roots of some trigonometric polynomials which depend explicitly on the side lengths and angles of the polygon. The proofs involve some classical hydrodynamics results related to a sloping beach problem, and to a sloshing problem. I’ll also state some open questions. The talk will be based on joint works with Leonid Parnovski, Iosif Polterovich, and David Sher, see arXiv:1908.06455 and arXiv:1709.01891. |
| Date: | October 11, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Irina Holmes, Texas A&M University |
| Title: | A new proof of the weak (1,1) inequality for the dyadic square function |
| Abstract: | This project (joint with Paata Ivanisvili and Sasha Volberg) is concerned with finding the (strange) sharp constant in the weak (1,1) inequality for the dyadic square function, using the Bellman function method. This constant was conjectured by Bollobas in the 1980’s and proved first by Osekowski using Brownian motion methods. The interesting aspect of our new proof is that it required the invention of a new way to work with Bellman functions - a way which we hope can be implemented in other problems. |
| Date: | October 18, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Cody Stockdale, Washington University, St. Louis |
| Title: | A Different Approach to Endpoint Weak-type Estimates for Calderón-Zygmund Operators |
| Abstract: | The weak-type (1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. This estimate was originally proved using the Calderón-Zygmund decomposition. To address more general settings, Nazarov, Treil, and Volberg gave a different proof of the weak-type (1,1) estimate. We investigate this alternative proof technique. We will compare the Calderón-Zygmund decomposition and Nazarov-Treil-Volberg techniques, discuss a simplification of the Nazarov-Treil-Volberg proof in the Lebesgue setting, and describe applications in a variation of the classical setting, weighted settings, and multilinear settings. |
| Date: | October 25, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | P. Kuchment, TAMU |
| Title: | On generic non-degeneracy of spectral edges. Discrete case |
| Abstract: | This is joint work with Frank Sottile (TAMU) and Ngoc T. Do (Missouri State U.) An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrodinger operator -Delta+V(x) in R^n with periodic potential near the edges of the spectrum, i.e. near extrema of the dispersion relation. A well known and widely believed conjecture says that generically (with respect to perturbations of the periodic potential) the extrema are attained by a single branch of the dispersion relation, are isolated, and have non-degenerate Hessian (i.e., dispersion relations are graphs of Morse functions). In particular, the important notion of effective masses hinges upon this property. The progress in proving this conjecture has been slow. It is natural to try to look at discrete problems, where the dispersion relation is (in appropriate coordinates) an algebraic, rather than analytic, variety. Moreover, such models are often used for computation in solid state physics (the tight binding model). Alas, counterexamples showing that the genericity can fail in some discrete situations do exist. In our work, we consider the case of a general periodic discrete operator depending polynomially on some parameters. We prove that the non-degeneracy of extrema either fails or holds for all but a proper algebraic subset of values of parameters. Thus, a random choice of a point in the parameter space will give the correct answer "with probability one". A specific example of a diatomic Z^2-periodic structure is also considered, which provides a cornucopia of examples for both alternatives, as well as a different approach to the genericity problem. |
| Date: | November 8, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Gregory Berkolaiko, Texas A&M University |
| Title: | Quantum graphs with a shrinking subgraph and exotic eigenvalues |
| Abstract: | We address the question of convergence of Schroedinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graph's edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. The failure is due to presence of what we call "exotic eigenvalues": eigenvalues whose eigenfunctions increasingly localize on the edges that are shrinking to a point. We establish a sufficient condition for convergence which encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges. In some important special cases this condition is also shown to be necessary. Moreover, when the condition fails, it provides quantitative information on exotic eigenvalues. Before formulating the main results we will review the setting of Schrodinger operators on metric graphs and the characterization of possible self-adjoint conditions, followed by numerous examples where the limiting operator is not obvious or where the convergence fails outright. The talk is based on a joint work with Yuri Latushkin and Selim Sukhtaiev, arXiv:1806.00561 (Adv. Math. 2019) and on work in progress with Yves Colin de Verdiere. |
| Date: | November 15, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Sohrab Shahshahani, UMass Amherst |
| Title: | Asymptotic stability of harmonic maps on the hyperbolic plane under the Schrodinger maps evolution |
| Abstract: | We consider the Cauchy problem for the Schrodinger maps evolution when the domain is the hyperbolic plane. An interesting feature of this problem compared to the more widely studied case on the Euclidean plane is the existence of a rich new family of finite energy harmonic maps. These are stationary solutions, and thus play an important role in the dynamics of Schrodinger maps. The main result is the asymptotic stability of (some of) such harmonic maps under the Schrodinger maps evolution. More precisely, we prove the nonlinear asymptotic stability of a finite energy equivariant harmonic map Q under the Schrodinger maps evolution with respect to non-equivariant perturbations, provided that Q obeys a suitable linearized stability condition. This is joint work with Andrew Lawrie, Jonas Luhrmann, and Sung-Jin Oh. |
| Date: | November 20, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 624 |
| Speaker: | Rodrigo Bezerra Matos, MSU |
| Title: | Localization in the disordered Hubbard model within Hartree-Fock theory |
| Abstract: | After a brief and self-contained review of the results on Anderson localization in the non-interacting setting, we shall discuss recent developments on the interacting context. This will be done for the disordered Hubbard model within the Hartree-Fock theory, which is an approximation used to understand qualitatively the time evolution of a particle subject not only to a random environment but also to infinitely many interactions. There, we prove (single particle) localization at any dimension in the regime of large disorder and at any disorder in the one-dimensional case. This is joint work with Jeffrey Schenker. |
| Date: | November 22, 2019 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Jonas Luhrmann, Texas A&M University |
| Title: | Stability of kinks and dispersive decay of Klein-Gordon waves |
| Abstract: | Kinks are particle-like solitons that arise in nonlinear scalar field theories in one space dimension. In this talk I will explain how the study of the asymptotic stability of kinks centers on the long-time behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient nonlinearities. Then I will present a new result on sharp decay estimates and asymptotics for small solutions to variable coefficient cubic nonlinear Klein-Gordon equations. If time permits, I will also discuss work in progress on the variable coefficient quadratic case, which exhibits a striking resonant interaction between the spatial oscillations of the variable coefficient and the temporal oscillations of the solutions. This is joint work with Hans Lindblad and Avy Soffer. |
