
Date Time 
Location  Speaker 
Title – click for abstract 

01/23 1:50pm 
BLOC 627 
Alfonso MontesRodriguez University de Seville, Spain 
Uncertainty Principle and Uniqueness Sets for the KleinGordon Equation


01/30 1:50pm 
BLOC 627 
Stephen Fulling Texas A&M University 
Two Apparent Paradoxes of Concrete Spectral Theory
Two recent papers in the physics literature claim results
that prima facie seem to be in conflict with the well established
theory of selfadjoint extensions and eigenfunction expansions
for ODEs. In one case an obviously limitpoint problem is
treated as if it were limitcircle: an "alternative" basis of
eigenfunctions is presented, with numerical verification. The
other paper finds a "continuous spectrum of normalizable
eigenfunctions", which probably indicates that a boundary
condition has been left out  i.e., a limitcircle problem is
being treated as limitpoint. Despite some mathematical naivete,
these papers do not exhibit glaring mistakes, and some of their
conclusions, properly interpreted, may be valuable insights that
would not be noticed by a mathematician following the traditional
approach. I report progress in resolving these two situations.
Input from experts in inverse problems and hyperbolic PDE would
help me to finish the job.


02/13 1:50pm 
BLOC 627 
Peter Kuchment Texas A&M University 
Parseval frames of exponentially decaying Wannier functions
When studying linear constant coefficient PDEs, or more generally
equations invariant with respect to shifts, the Fourier transform
is known to be very useful, as well as two basic sets of functions:
plane waves and deltafunctions, which are interchanged by the Fourier
transform.
In case of equations invariant with respect to a lattice
in R^{n} only (e.g., the solid state physics' Schroedinger equation
with periodic potential or photonic crystal theory's Maxwell
operator in a periodic dielectric medium), the Fourier transform and the
corresponding function sets are not useful anymore. However, analogs of
these are known: FloquetBloch transform replaces the Fourier one, Bloch
functions play the role of plane waves, and the so called Wannier
functions are analogs of delta functions. All of these are heavily used
in physics and mathematics. The problem is with the localization of Wannier
functions: one wants them to decay as fast as possible (exponentially is
prefered); after all, they are analogs of delta functions! And here one
hits a snag: it has been known since 1980s (Thoules) that existence of an
orthonormal basis of exponentially (in fact, even much slower) decaying
Wannier functions faces a topological obstruction in the form of possible
nontriviality of a vector bundle on a torus. Efforts are still being
spent on finding cases where the bundle is trivial, although generically
(e.g., in magnetic fields) it is not.
In this talk I will present a recent result showing that if one relaxes
the orthonormal basis condition and asks only for a Parseval (analog of
orhtonormality) frame of such functions, it always exists. The bundle
tells one how redundant the frame should be (if the bundle is trivial, the
frame is a basis).
The techniques in proving this result come from spectral theory, several
complex variables and some elementary algebraic topology.
No knowledge of these topics will be assumed, though :) 

03/06 1:50pm 
BLOC 627 
E. Abakumov Univerity of MarnelaVallee 
Approximation by translates in weighted spaces
We discuss some problems on approximation by discrete translates
of a given function in weighted function spaces.


03/13 3:00pm 
BLOC 627 
Dmitry Panchenko Texas A&M University 
The GhirlandaGuerra identities and ultrametricity in the SherringtonKirkpatrick model
The Parisi theory of the SK model completely describes the geometry of the Gibbs sample in a sense that it predicts the limiting joint distribution of all scalar products, or overlaps, between i.i.d. replicas. One of the main properties of this distribution is the ultrametricity which means that the Gibbs
measure approximately concentrates on the ultrametric subset of all configurations; another property is the GhirlandaGuerra distributional identities. It is well known that these two properties completely determine the distribution and, probably for this reason, they were always considered complementary. We show that if an overlap takes finitely many values then the GhirlandaGuerra identities actually imply ultrametricity.


03/27 1:50pm 
BLOC 627 
Misko Mitkovski Texas A&M University 
Polya sets, gap theorems and Toeplitz kernels.
A separated sequence of real numbers is called a Polya sequence if any entire
function of exponential type zero that is bounded on that sequence is a constant. We give a description of all such sequences thus solving a problem of Polya and Levinson. Furthermore, we show that this problem is equivalent to a version of a Beurling gap problem. The key step is to give a description of both problems in terms of kernels of certain Toeplitz operators. Finally, we use Toeplitz kernels versions of BeurlingMalliavin theorems given recently by Makarov and Poltoratski to obtain the complete metric description.


04/03 3:00pm 
BLOC 627 
Joel Zinn Texas A&M University 
On the Gaussian Correlation Inequality


04/08 4:00pm 
BLOC 627 
O. Post Humbolt University, Berlin 
Quantum graph approximations of thin branching structures
Many physical systems have branching structure of thin transversal diameter. One can name for instance quantum wire circuits, thin branching waveguides, or carbon nanostructures. In applications, such systems are often approximated by the underlying onedimensional graph structure, a socalled "quantum graph". In this way, many properties of the system like conductance can be calculated easier (sometimes even explicitly). After briefly explaining the notion of a quantum graph, we show that the system with thin transversal diameter converges to a quantum graph. We also identify which vertex couplings (which influence the current through the graph) can be obtained by appropriate engineering of the branching structure. 

04/17 1:50pm 
BLOC 627 
D. Damanik Rice University 
Pseudorandom Potentials: Open Problems and Some Recent Results
A bounded infinite sequence of real numbers is called a pseudorandom potential if the Schrodinger operator with this sequence as a potential has spectral properties akin to those of a typical Schr"odinger operator with a random sequence as a potential. We present several sequences that are conjectured to be pseudorandom potentials based on heuristics and numerics and then explain some related rigorous results.


04/24 1:50pm 
BLOC 627 
R. Band Weizmann Institute of Science 
Counting nodal domains on quantum graphs
The investigation of nodal domains on manifolds has began already in the 19th century by the pioneering work of Chladni on the nodal structures of vibrating plates. Counting nodal domains started with Sturm's oscillation theorem which states that a vibrating string is divided into exactly n nodal intervals by the zeros of its nth vibrational mode. A quantum graph can be thought of as a structure of strings which are attached to each other. For a given quantum graph the nodal count sequence is the sequence of numbers of nodal intervals of its vibrational modes ordered by their frequencies. Many recent works treated the bounds and the statistics of the nodal sequences both for graphs and for manifolds. Nevertheless, an exact formula for the nodal sequence is still not available. We show the existence of a formula for a specific quantum graph and offer an approach which might yield nodal count formulae for quantum graphs.
Such a formula would help in answering the inverse question regarding the geometrical information that is stored in the nodal sequence. This is a joint work with Gregory Berkolaiko and Uzy Smilansky. 

05/01 1:50pm 
BLOC 627 
H. Krueger Rice University 
The potential V(n) = f(n^{ρ} (mod 1))
Consider the potential V(n) = f(n^{ρ} (mod 1)) for ρ > 0 not an integer, and the associated Schrodinger operator H. I will study the spectrum and Lyapunov exponent of this operator, in particular deriving explicit formulas for it in the case 0 < ρ < 1.


05/06 4:15pm 
BLOC 627 
Y. Lyubarskii NTNU, Norway 
Direct and Inverse Problem of Multichannel scattering
We consider direct and inverse scattering problems for a system of particles interacting pairwise with each other and perhaps with an external field. It is assumed that the system consists of a finite set of channels, i.e., semiinfinite chains of particles, attached to an "innerpart"  a finite system of interacting particles, and that the system performs small oscillations around stable equilibrium. We show that having the scattering data one can reconstruct the characteristics of the system along the channels and also find conditions which allow to reconstruct the characteristics of the inner part of the system.
This is a joint work with Vladimir Marchenko (Institute for Low Temperatures, Ukraine)


05/14 1:00pm 
BLOC 627 
V. Gurarii Swinburne University of Technology 
Divergent expansions: Error bounds, Duality and Stokes phenomenon
We consider classes of functions uniquely determined by coefficients of their divergent expansions. By approximating a function in such a class by partial sums of its expansion we study how accuracy changes when we move within a region of the complex plane. This enables us to discover some features of Stokes phenomenon which are not covered in the literature and to propose a theory of divergent expansions, which includes Stokes phenomenon as its essential part. This in turn allows us to formulate necessary and sufficient conditions for divergent expansions to encounter Stokes' phenomenon. 