MenuMathematical Physics and Harmonic Analysis Seminar
Spring 2018
| Date: | February 9, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | James Kennedy, University of Lisbon |
| Title: | Asymptotically optimal Laplacian eigenvalues and Polya's conjecture |
| Abstract: | A longstanding problem in spectral geometry is to determine the domain(s) which minimise a given eigenvalue of a differential operator such as the Laplacian with Dirichlet boundary conditions, among all domains of given volume. For example, the Theorem of (Rayleigh--) Faber--Krahn states that the smallest eigenvalue is minimal when the domain is a ball. Very little to nothing is known about domains minimising the higher eigenvalues, but the Weyl asymptotics suggest that the ball should in a certain sense be asymptotically optimal. In the first part of this talk, we will sketch a new approach to this problem initiated by a paper of Colbois and El Soufi in 2014, which asks not after the minimising domains themselves but properties of the corresponding sequence of minimal values. This serendipitously also yields a new Ansatz for tackling the more than 50 year old conjecture of Polya that the k-th eigenvalue of the Dirichlet Laplacian on any domain always lies above the corresponding first term in the Weyl asymptotics for that eigenvalue. Along the way, we will additionally meet variants of Gauss circle problem. In a second part, we will present some recent analogous results for the Laplacian with Robin boundary conditions, which are ongoing joint work with Pedro Freitas. |
| Date: | February 9, 2018 |
| Time: | 2:50pm |
| Location: | Blocker 605AX |
| Speaker: | Yaiza Canzani, UNC Chapel Hill |
| Title: | On the growth of eigenfunctions averages |
| Abstract: | In this talk we discuss the behavior of Laplace eigenfunctions when restricted to a fixed submanifold by studying the averages given by the integral of the eigenfunctions over the submanifold. In particular, we show that the averages decay to zero when working on a surface with Anosov geodesic flow regardless of the submanifold (curve) that one picks. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages. This is based on joint work with Jeffrey Galkowski. |
| Date: | February 23, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | George E. A. Matsas, Instituto de Fisica Teorica, Universidade Estadual Paulista |
| Title: | Overview of the Unruh Effect for Mathematicians |
| Abstract: | The Unruh effect is interesting to physicists and mathematicians. Unveiled by a physicist, Bill Unruh, in 1975, it vindicated Steve Fulling's surprising conclusion that different observers extract, in general, different particle contents from the same field theory (e.g., inertial observers in the usual vacuum would freeze to death at 0 K, where observers accelerated enough may burn into ashes). This seminar is designed for mathematicians who are not acquainted with quantum field theory but wish to understand what the Unruh effect means, up to what extent we must trust it, and why it is so important to our comprehension of some conceptual issues. |
| Date: | March 2, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Gregory Berkolaiko, TAMU |
| Title: | Nodal count distribution of graph eigenfunctions |
| Abstract: | We start by reviewing the notion of “quantum graph”, its eigenfunctions and the problem of counting the number of their zeros. The nodal surplus of the n-th eigenfunction is defined as the number of its zeros minus (n-1). When the graph is composed of two or more blocks separated by bridges, we propose a way to define a “local nodal surplus” of a given block. Since the eigenfunction index n has no local meaning, the local nodal surplus has to be defined in an indirect way via the nodal-magnetic theorem of Berkolaiko, Colin de Verdière and Weyand. We will discuss the properties of the local nodal surplus and their consequences. In particular, its symmetry properties allow us to prove the long-standing conjecture that the nodal surplus distribution for graphs with β disjoint loops is binomial with parameters (β,1/2). The talk is based on joint work with Lior Alon and Ram Band, arXiv:1709.10413 (accepted to CMP). |
| Date: | March 9, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Peter Kuchment, Texas A&M |
| Title: | On Liouville-Riemann-Roch theorems on co-compact abelian coverings |
| Abstract: | A generalization by Gromov and Shubin [2-3] of the classical Riemann-Roch theorem describes the index of an elliptic operator on a compact manifold with a divisor of prescribed zeros and allowed singularities. On the other hand, Liouville type theorems count the number of solutions of a given polynomial growth of the Laplace-Beltrami (or more general elliptic) equation on a non-compact manifold. The solution of a 1975 Yau's conjecture [6] by Colding and Minicozzi [1] implies in particular, that such dimensions are finite for Laplace-Beltrami equation on a nilpotent co-compact covering. In the case of an abelian covering, much more complete Liouville theorems (including exact formulas for dimensions) have been obtained by Kuchment and Pinchover [4-5]. One wonders whether such results have a combined generalization that would allow for a divisor that "includes the infinity." Surprisingly, combining the two types of results turns out being rather non-trivial. The talk will present such a result obtained recently in a joint work with Minh Kha (former A&M PhD student, currently postdoc at U. Arizona). [1] Colding, T. H., Minicozzi, W. P.: Harmonic functions on manifolds, Ann. of Math. 146 (1997), 725–747. [2] M. Gromov and M. A. Shubin, The Riemann-Roch theorem for elliptic operators, I. M. Gel'fand Seminar, 1993, pp. 211--241. [3] --" -- , The Riemann-Roch theorem for elliptic operators and solvability of elliptic equations with additional conditions on compact subsets, Invent. Math. 117 (1994), no. 1, 165--180. [4] P. Kuchment and Y. Pinchover, Integral representations and Liouville theorems for solutions of periodic elliptic equations, J. Funct. Anal. 181 (2001), no. 2, 402--446. [5] --"-- , Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5777--5815. [6] Yau, S. T.: Harmonic functions on complete Riemannian manifolds, Commun. Pure Appl. Math. 28 (1975), 201–228. |
| Date: | April 6, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Oran Gannot, Northwestern University |
| Title: | Semiclassical diffraction by conormal potential singularities |
| Abstract: | I will describe joint work with Jared Wunsch on propagation of singularities for some semiclassical Schrödinger equations, where the potential is conormal to a hypersurface. Semiclassical singularities of a given strength propagate across the hypersurface up to a threshold depending on both the regularity of the potential and the singularities along certainbackwards broken bicharacteristics. |
| Date: | April 13, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Vitaly Moroz, University of Swansea |
| Title: | Asymptotic properties of ground states of a semilinear elliptic problem with a vanishing parameter. |
| Abstract: | We consider an elliptic problem with a double-well nonlinearity and a vanishing parameter. The behaviour of solutions depends sensitively on whether a power in the nonlinearity is less, equal or bigger than the critical Sobolev exponent. In the most delicate critical Sobolev regime the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden-Fowler equation, whose choice depends on in a non-trivial way on the space dimension. Joint work with Cyrill Muratov (NJIT). |
| Date: | April 16, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 220 |
| Speaker: | Barry Simon , Caltech |
| Title: | Szegő–Widom asymptotics for Chebyshev polynomials on subsets of R |
| Abstract: | Chebyshev polynomials for a compact subset e ⊂ R are defined to be the monic polynomials with minimal $||·||_∞ $ over e. In 1969, Widom made a conjecture about the asymptotics of these polynomials when e was a finite gap set. We prove this conjecture and extend it also to those infinite gap sets which obey a Parreau–Widom and a Direct Cauchy Theory condition. This talk will begin with a generalities about Chebyshev Polynomials. This is joint work with Jacob Christiansen and Maxim Zinchenko and partly with Peter Yuditskii. |
| Date: | April 16, 2018 |
| Time: | 4:00pm |
| Location: | BLOC 117 |
| Speaker: | Barry Simon , Caltech |
| Title: | A colloquium talk: Tales of our Forefathers |
| Abstract: | This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. |
| Date: | April 20, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | David Borthwick, Emory University |
| Title: | Distribution of Resonances for Hyperbolic Surfaces |
| Abstract: | For non-compact hyperbolic surfaces, the appropriate generalization of the eigenvalue spectrum is the resonance set, the set of poles of the resolvent of a meromoprhic continuation of the Laplacian. Hyperbolic surfaces serve as a model case for quantum theory when the underlying classical dynamics is chaotic. In this talk I’ll explain how the resonances are defined and discuss our current understanding of their distribution. I’ll introduce some conjectures inspired by the physics of quantum chaotic systems, and discuss numerical evidence for these conjectures and the partial progress that has been made recently. |
| Date: | May 4, 2018 |
| Time: | 1:50pm |
| Location: | BLOC 628 |
| Speaker: | Junehyuk Jung, Texas A&M University |
| Title: | Boundedness of the number of nodal domains of eigenfunctions |
| Abstract: | The asymptotic of the number of nodal domains of eigenfunctions on a manifold is closely related to the dynamics of the geodesic flow on the manifold. For instance, if a surface with non-empty boundary has an ergodic geodesic flow, then for any given Dirichlet eigenbasis, one can find a subsequence of density one where the number of nodal domains tends to +\infty. In this talk, I'm going to discuss what happens to the unit circle bundle over a manifold. When equipped with a metric which makes the Laplacian to commute with the circular action on each fiber, the geodesic flow never is ergodic. Recently I and Steve Zelditch proved that among such metrics the following property is generic: One can find an eigenbasis that has a subsequence of density 1 where the number of nodal domains is identically 2. This highlights how underlying dynamics can impact the nodal counting. I will sketch proof when we are considering a unit tangent bundle of a compact surface with the genus \neq 1. |
