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.

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.

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.

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.

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.

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.