The classical Noether-Lefschets Theorem states that for a sufficiently general surface S in P^3 the only algebraic curves lying on S are the complete intersections. In 2010 we proved an extension of this result to surfaces (and higher dimensional hypersurfaces in P^n) containing a fixed base locus. I will discuss this result and the describe how it can be applied together with tools from complex geometry and formal power series, to the study of class groups of local rings, in particular how they vary within an analytic isomorphism class. Among other things, we prove that any hypersurface singularity is analytically isomorphic to one whose local ring is a UFD and give a complete classification of the possible class groups for rational double point surface singularities.

I will present low-order models for fluid-structure interactions in fish locomotion and comment on both active (controlled) and passive (open-loop) dynamics and stability. I will also discuss a finite dipole model. The motivation for the latter is to develop scalable models for fish schooling that account for the role of hydrodynamic coupling among the fish in a school. The finite dipole model exhibits interesting dynamics. I conclude by commenting on the advantages and limitations of such low-order modeling approach.

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.