
Date Time 
Location  Speaker 
Title – click for abstract 

08/30 1:50pm 
BLOC 628 
P. Kuchment 
CANCELLED
due to the funeral of Professor Michael Boshernitzan (Rice U.)
Analyticity of the spectrum and DirichlettoNeumann 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 DirichlettoNeumann (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) 

09/06 1:50pm 
BLOC 628 
Robert Booth Texas A&M University 
Almost Global Existence for Asymptotically Euclidean Quasilinear Wave Equations
In this talk, we will discuss a recent result demonstrating almost global existence for a class of nontrapping 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. 

09/12 2:50pm 
BLOC 220 
Burak Hatinoglu TAMU 
A complex analytic approach to mixed spectral problems (Unusual day, time and room)
We consider a generalization of classical inverse spectral results of Borg and Marchenko for the Schroedinger operators on a finite interval with an L^1potential. 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?


09/13 1:50pm 
BLOC 628 
Selim Sukhtaiev Rice University 
Hadamardtype formulas and resolvent expansions
In this talk I will discuss an Hadamardtype variation formula for eigenvalue curves of oneparameter families of selfadjoint 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. 

09/20 1:50pm 
BLOC 628 
Junho Yang Texas A&M University, Statistics 
Rate of convergence in Szego limit theorems
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 pointwise asymptotics of all eigenvalues of banded Hermitian matrices; 2) Approximate general nbyn Toeplitz matrix with pdiagonal banded Hermitian matrix in the sense of the "best predictive" function, and provide an entrywise 1norm 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). 

09/25 1:50pm 
BLOC 628 
Lior Alon Technion  Israel Institute of Technology 
On a CLT conjecture for the nodal statistics of quantum graphs
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. 

09/27 1:50pm 
BLOC 628 
Blake Keeler UNC 
Random Waves and the Spectral Function on Manifolds without Conjugate Points
In this talk, we discuss offdiagonal 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 LaplaceBeltrami operator. We will prove that in the geometric setting of manifolds without conjugate points, one can obtain a logarithmic improvement in the twopoint 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 ondiagonal case for manifolds with nonpositive curvature. 

10/04 1:50pm 
BLOC 628 
Wencai Liu Texas A&M University 
Anderson localization for multifrequency quasiperiodic operators on higher dimensional latices
The first part of the talk, based on a joint work with S. Jitomirskaya and Y. Shi, is devoted to study
multifrequency quasiperiodic operators on higher dimensional lattices. We establish
the Anderson localization for general analytic $k$frequency quasiperiodic
operators on $\Z^d$ for arbitrary $k, d$.
This is a generalization of BourgainGoldsteinSchlag'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 (KolmogorovArnoldMoser) tori for NLS and NLW equations.


10/09 1:50pm 
BLOC 624 
Michael Levitin University of Reading (UK) 
Asymptotics of Steklov eigenvalues for curvilinear polygons (Unusual date and room!)
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. 

10/11 1:50pm 
BLOC 628 
Irina Holmes Texas A&M University 
A new proof of the weak (1,1) inequality for the dyadic square function
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. 

10/18 1:50pm 
BLOC 628 
Cody Stockdale Washington University, St. Louis 
A Different Approach to Endpoint Weaktype Estimates for CalderónZygmund Operators
The weaktype (1,1) estimate for CalderónZygmund operators is fundamental in harmonic analysis. This estimate was originally proved using the CalderónZygmund decomposition. To address more general settings, Nazarov, Treil, and Volberg gave a different proof of the weaktype (1,1) estimate. We investigate this alternative proof technique. We will compare the CalderónZygmund decomposition and NazarovTreilVolberg techniques, discuss a simplification of the NazarovTreilVolberg proof in the Lebesgue setting, and describe applications in a variation of the classical setting, weighted settings, and multilinear settings. 

10/25 1:50pm 
BLOC 628 
P. Kuchment TAMU 
On generic nondegeneracy of spectral edges. Discrete case
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 nondegenerate 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 nondegeneracy 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^2periodic structure is also considered, which provides a cornucopia of examples for both alternatives, as well as a different approach to the genericity problem. 

11/08 1:50pm 
BLOC 628 
Gregory Berkolaiko Texas A&M University 
Quantum graphs with a shrinking subgraph and exotic eigenvalues
We address the question of convergence of Schroedinger operators on metric graphs with general selfadjoint 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 Sobolevtype 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 selfadjoint 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. 

11/15 1:50pm 
BLOC 628 
Sohrab Shahshahani UMass Amherst 
Asymptotic stability of harmonic maps on the hyperbolic plane under the Schrodinger maps evolution
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
nonequivariant perturbations, provided that Q obeys a suitable linearized
stability condition. This is joint work with Andrew Lawrie, Jonas
Luhrmann, and SungJin Oh. 

11/20 1:50pm 
BLOC 624 
Rodrigo Bezerra Matos MSU 
Localization in the disordered Hubbard model within HartreeFock theory
After a brief and selfcontained review of the results on Anderson localization in the noninteracting setting, we shall discuss recent developments on the interacting context. This will be done for the disordered Hubbard model within the HartreeFock 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 onedimensional case. This is joint work with Jeffrey Schenker. 

11/22 1:50pm 
BLOC 628 
Jonas Luhrmann Texas A&M University 
Stability of kinks and dispersive decay of KleinGordon waves
Kinks are particlelike 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 longtime behavior of small solutions to onedimensional KleinGordon 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 KleinGordon 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.
