There are many recent points of contact of model theory and other
parts of mathematics: o-minimality and Diophantine geometry, geometric group
theory, additive combinatorics, rigid geometry,... I will probably
emphasize long-standing themes around stability, Diophantine geometry, and
analogies between ODE's and bimeromorphic geometry.
Many geophysical flows over topography can be modeled by two-dimensional
depth-averaged fluid dynamics equations. The shallow water equations
are the simplest example of this type, and are often sufficiently
accurate for simulating tsunamis and other large-scale flows such
as storm surge. These hyperbolic partial differential equations
can be modeled using high-resolution finite volume methods. However,
several features of these flows lead to new algorithmic challenges,
e.g. the need for well-balanced methods to capture small perturbations
to the ocean at rest, the desire to model inundation and flooding,
and that vastly differing spatial scales that must often be modeled,
making adaptive mesh refinement essential. I will discuss some of
the algorithms implemented in the open source software GeoClaw that
is aimed at solving real-world geophysical flow problems over
topography. I'll also show results of some recent studies of the
11 March 2011 Tohoku Tsunami and discuss the use of tsunami modeling
in probabilistic hazard assessment.
Yves Couder and co-workers have recently reported the results of a startling series of experiments in which droplets bouncing on a fluid surface exhibit several dynamical features previously thought to be peculiar to the microscopic realm. In an attempt to
develop a connection between the fluid and quantum systems, we explore the Madelung transformation, whereby Schrodinger's equation is recast in a hydrodynamic form. New experiments are presented, and indicate the potential value of this hydrodynamic approach to both visualizing and understanding quantum mechanics.
Voiculescu showed how the large N limit of the expected value of the trace of a word on n independent hermitian NxN matrices gives a well known von Neumann algebra. In joint work with Guionnet and Shlyakhtenko it was shown that this idea makes sense in the context of very general planar algebras where one works directly in the large N limit. This allowed us to define matrix models with a non-integral number of random matrices. I will present this work and some of the subsequent work, together with future hopes for the theory.