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.

Basic facts on towers and homogeneity systems will be presented.

In the talk we address a system of PDEs describing an
interaction between an incompressible fluid and an elastic
body. The fluid motion is modeled by the Navier-Stokes
equations while an elastic body evolves according to an
linear elasticity equation. On the common boundary, the
velocities and stresses are matched. We discuss available
results on local well-posedness and prove new existence and
uniqueness results with the velocity and the displacement
belonging to low regularity spaces.

The results are joint with A. Tuffaha.

A smooth metric space is a Riemanian manifold together with a weighted volume. It is naturally associated with a weighted Laplacian. In this talk, I will discuss some recent results about function theoretic and spectral propeties of the weighted Laplacian and volume estimates for the volume and weighted volume. The results can be applied to study the shrinking gradient Ricci solitons and self-shrinker for mean curvature flows.