A famous theorem of Wiener states that if a periodic function has an absolutely convergent Fourier series and never vanishes, then its reciprocal also has an absolutely convergent Fourier series. In a forthcoming paper by K. Grochenig and M. Leinert (Journal of the American Mathematical Society), this is generalized, using the techniques of abstract harmonic analysis to a noncommutative setting, and then applied to the theory of Gabor frames in time-frequency analysis. In this talk,
I'll present their proof of the generalization. Future talks will be devoted to the applications mentioned above.

We discuss the trace map associated with the Fibonacci
quasicrystal. While the associated dynamical system has been studied heavily as a real dynamical system, it may also be regarded as a complex dynamical system. We study the stable set and give explicit bounds for the complex approximants. Quantum dynamical consequences of these results will be
explained. This is joint work with Serguei Tcheremchantsev.

The NLS equation describes solitonic transmission in
fiber optic communication and is generically encountered in propagation through nonlinear media. One of its most important aspects is its modulational instability: regular wavetrains are unstable to modulation and break up to more complicated structures.

The IVP for the NLS equation is solvable by the method of inverse scattering. The initial spectral data of the Zakharov Shabhat (ZS) operator, a particular linear operator having the solution to NLS as the potential, are calculated from the initial data of the NLS; they evolve in a simple way as a result of the integrability of the problem, and produce the solution through the inverse spectral transformation.

In collaboration with A. Tovbis, we have developed
a one parameter family of initial data for which the derivation of the spectral data is explicit. Then, in collaboration with A. Tovbis and X. Zhou, we have obtained the following results:

1) We prove the existence and basic properties of the
first breaking curve (curve in space-time above which the character
of the solution changes by the emergence of a new
oscillatory phase) and show that for pure radiation
no further breaks occur.

2) We construct the solution beyond the first break-time.

3) We derive a rigorous estimate of the error.

4) We derive rigorous asymptotics for the large
time behavior of the system in the pure radiation case.

The geometry of Sp(3)/U(3) as a subvariety of Gr(3,6) will be explored to explain several examples given by Mukai of non-abelian Brill-Noether loci, and to give some new examples. These examples identify Brill-Noether loci of vector bundles on linear sections of the Lagrangian Grassmannian Sp(3)/U(3) with orthogonal linear sections of the dual variety and vice versa. We will show that any nodal hyperplane section of the Lagrangian Grassmannian projected from the node is a linear section of the Grassmannian Gr(2,6).

We consider paraorthogonal polynomials P_n on the unit circle defined by
random recurrence (Verblunsky) coefficients. Their zeros are exactly
the eigenvalues of a special class of random unitary matrices (random CMV
matrices). We prove that the local statistical distribution of these zeros
converges to a Poisson distribution. This means that, for large n, there
is no local correlation between the zeros of the random polynomials P_n.