We discuss how BMM affects the large cardinal
structure of V as well as the size of \theta^{L(R)}. BMM proves
that V is closed under sharps (and more), and BMM plus the
existence of a precipitous ideal on \omega_1 proves that
\delta^1_2 = \aleph_2. Part of this is joint work with my
student Ben Claverie.

The talk deals with the spectral analysis of Jacobi matrices superimposed
with random perturbations that decay in a certain sense.

We shall focus our attention on two problems: The first is the analysis of
spectral stability. We show that the absolutely continuous spectrum
associated with bounded generalized eigenfunctions, for Jacobi matrices with
a mild growth restriction on the off-diagonal terms, is stable under random
Hilbert-Schmidt perturbations. We also give some results for singular
spectral types. This is joint work with Yoram Last.

The second problem is the spectral analysis of Jacobi matrices arising in
the study of Gaussian \beta ensembles of Random Matrix Theory. These
matrices may be viewed as simple Jacobi matrices (with growing off-diagonal
terms) with a random perturbation that decays in a certain sense. With the
help of the appropriately modified methods, we analyze the behavior of the
generalized eigenfunctions and the Hausdorff dimension of the spectral
measure. Some of this work is joint with Peter Forrester and Uzy Smilansky.

A non-zero Abelian differential in a compact Riemann surface of genus $g \geq 1$ endows the surface with an atlas (outside the zeroes) whose coordinate changes are translations. There is a natural ``vertical flow'' (moving up with unit speed) associated with the translation structure, generalizing the genus $1$ case of irrational flows on the torus.

The Teichm\"uller flow in the moduli space of Abelian differentials can be seen as the renormalization operator of translation flows. In this talk, we will discuss how the chaoticity of the Teichm\"uller flow dynamics reflects on the (non-chaotic) dynamics of the associated vertical flows (for typical parameters), and the closely related interval exchange transformations.

Classical ideal fluid motion is described by Euler and Navier-Stokes equations. For real fluids, their motions are more complicated and governed by Euler and Navier-Stokes equations coupled with various constitutive equations. We study viscoelastic models whose motions are
carried out by the competition between the kinetic energies and internal elastic energies. The deformation tensor plays an essential role in our studies. We will present how to use the heuristics coming from the special
structure of the deformation tensor to establish the global well-posedness results for several viscoelastic models, but will focus on a 2D Strain-Rotation model.