Georgia Institute of Technology College of Sciences

We will actually finish our proof of the key technical lemma for the quantitative unique continuation principle of Ding-Smart, reviewing briefly the volumetric bound from the theory of \varepsilon-coverings/nets/packings. From there, we will outline at a high level the strategy for the rest of the proof of the unique continuation principle using this key lemma, before starting the proof in earnest.

The general subject of the talk is spectral theory of discrete (tight-binding) Schrodinger operators on d-dimensional lattices. For operators with periodic potentials, it is known that the spectra of such operators are purely absolutely continuous. For random i.i.d. potentials, such as the Anderson model, it is expected and can be proved in many cases that the spectra are almost surely purely point with exponentially decaying eigenfunctions (Anderson local- ization). Quasiperiodic operators can be placed somewhere in between: depending on the potential sampling function and the Diophantine properties of the frequency and the phase, one can have a large variety of spectral types. We will consider quasiperiodic operators

(H(x)ψ)n =ε(∆ψ)n +f(x+n·ω)ψn, n∈Zd,

where ∆ is the discrete Laplacian, ω is a vector with rationally independent components, and f is a 1-periodic function on R, monotone on (0,1) with a positive lower bound on the derivative and some additional regularity properties. We will focus on two methods of proving Anderson localization for such operators: a perturbative method based on direct analysis of cancellations in the Rayleigh-Schr ̈odinger perturbation series for arbitrary d, and a non?perturbative method based on the analysis of Green?s functions for d = 1, originally developed by S. Jitomirskaya for the almost Mathieu operator.

The talk is based on joint works with S. Krymskii, L. Parnovski, and R. Shterenberg (per- turbative methods) and S. Jitomirskaya (non-perturbative methods).

We will consider the inverse problem of determining the sound
speed or index of refraction of a medium by measuring the travel times of
waves going through the medium. This problem arises in global seismology
in an attempt to determine the inner structure of the Earth by measuring
travel times of earthquakes. It also has several applications in optics
and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine
the Riemannian metric of a Riemannian manifold with boundary by
measuring the distance function between boundary points? This is the
boundary rigidity problem.

We will also describe some recent results, joint with Plamen Stefanov
and Andras Vasy, on the partial data case, where you are making
measurements on a subset of the boundary.

We will look at a number of interesting examples — some proven, others merely conjectured — of Hamburger moment sequences in combinatorics. We will consider ways in which this positivity may be expected, for instance in different types of combinatorial statistics on perfect matchings that turn out to encode moments in noncommutative analogues of the classical Central Limit Theorem. We will also consider situations in which this positivity may be surprising, and where proving it would open up new approaches to a class of very hard open problems in combinatorics.