We continue the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We establish some relations with forcing axioms and with the existence of certain regular forcing embeddings, and also point out connections with Precipitousness. In particular we observe that if Projective Catch holds for an ideal, then that ideal is precipitous, and the converse holds for ideals that concentrate on countable sets. Finally we give an overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory.

Two very different mathematical topics, where one comes from the social sciences and the other from the physical sciences.  It is easy to expect that they have nothing to do with one another.  Instead, and as it will be developed in this talk,  thanks to connecting power of  mathematics, they share several relationships.

 In this talk, I will discuss recent results on the
 large time well-posedness of classical solutions to the
 multi-dimensional compressible Navier-Stokes system with possible
 large oscillations and vacuum.
 The focus will be on finite-time blow-up of classical solutions for
 the 3-D full compressible Navier-Stokes system, and the global
 existence of classical solutions to the isentropic compressible
 Navier-Stokes system in both 2-D and 3-D in the presence of vacuum
 and possible large oscillations.  New estimates on the fast decay
 of the pressure in the presence of vacuum will be presented,  which
 are crucial for the well-posedness theory in 2-dimensional case.

We introduce a new multiscale Gaussian beam method for the
 numerical solution of the wave equation with smooth variable
 coefficients. The first computational question addressed in this
 paper is how to generate a Gaussian beam representation from general
 initial conditions for the wave equation. We propose fast multiscale
 Gaussian wavepacket transforms and introduce a highly efficient
 algorithm for generating the multiscale beam representation for a
 general initial condition. Starting from this multiscale
 decomposition of initial data, we propose the multiscale Gaussian
 beam method for solving the wave equation. The second question is
 how to perform long time propagation. Based on this new
 initialization algorithm, we utilize a simple reinitialization
 procedure that regenerates the beam representation when the beams
 become too wide. Numerical results in one, two, and three dimensions
 illustrate the properties of the proposed algorithm. The methodology
 can be readily generalized to treat other wave propagation problems.

Advisor: Professor Jack Xin