We use the Lippmann Schwinger equations to derive a relation between the transfer and
the scattering matrix for a quasi one-dimensonal scattering problem with a periodic background
operator.
If the background operator has hyperbolic channels, then the scattering matrix is of smaller
dimension than the transfer matrix and related to a 'reduced' transfer matrix.

The continuity of Lyapunov exponent plays an important role for many problems in quasi-periodic cocycles. One example is Ten Martini problem. It is well known that the Lyapunov exponent is continuous in analytic topology and discontinuous in C^0-topology. In this talk, we will provide quasi-periodic cocycles at which the Lyapunov exponent is not continuous in C^l-topology with 0 ≤ l ≤ +∞. This is joint work with Jiangong You.

Discrete quasiperiodic Schrodinger operators have been researched extensively over the past thirty years to produce a rather complete spectral analysis when the potential is defined by analytic functions. However, the nature of the spectral measures for less than $C^\infty$ regularity of the potential is largely unknown. We demonstrate that, with only minimal assumptions on the regularity of the potential, in the regime of positive Lyapunov exponents, the spectral measures are always of
Hausdorff dimension zero.

In physics, the main objective in spin glasses is to understand
the strange magnetic properties of alloys.
Yet the models invented to explain the observed phenomena are of a rather
fundamental nature in mathematics. In this talk, we will focus on one of the most important mean field
models,
called the Sherrington-Kirkpatrick model,
and discuss its disorder chaos problem. Using Guerra's replica
symmetric-breaking bound, we present a mathematically rigorous proof for this problem.

The talk will review some elementary, but amusing, results concerning
surface spectra for discrete Laplacians on half-planes with a boundary.In particular, interesting differences arise for square lattices with
straight boundaries between the case where the boundary has the same
direction of the lattice and the one where the boundary is slanted
at an angle of 45 degrees to the direction of the lattice. This
is joint work with Y. Kreimer.