December 13, 2019

Applied Math Colloquium Series 2002-2003


January 10, 2003
Joel Smoller, University of Michigan

Cosmology, Black Holes, and Shock Waves Beyond the Hubble Distance
We construct a class of global exact solutions of the Einstein equations that extend the Oppenheimer-Snyder (OS) model to the case of non-zero pressure, "inside a black-hole", by incorporating a shock wave at the leading edge of the expansion of the galaxies, arbitrarily far beyond the Hubble length in the Friedman-Robertson-Walker (FRW) spacetime. Here the expanding FRW universe emerges behind a subluminous blast wave that explodes outward from the FRW center at the instant of the Big Bang. The equation of state p=(1/3)(rho) plays a special role, and only in this case, the shock wave emerges from the Big Bang at the speed of light, decelerating from that time onward. The entropy condition implies that the shock wave must weaken to the point where it settles down to an OS interface, that eventually emerges from the White Hole event horizon of an ambient Schwarzschild spacetime. The entropy condition also breaks the time symmetry of the Einstein equations, selecting the explosion over the implosion. These shock wave solutions indicate a new cosmological model in which the Big Bang arises from a localized exlosion occurring inside the Black hole of an asymptotically flat Schwarzschild spacetime. (This is joint work with Blake Temple.) I will strive to make this talk understandable to non-experts.

February 17, 2003
Rafe Mazzeo, Stanford University
Poincare-Einstein metrics on the large and small scale
Poincar\'e-Einstein (also known as asymptotically hyperbolic Einstein) metrics have received substantial recent attention from both Riemannian geometers and string theorists. In this talk I will begin by reviewing the general theory and significant results about these spaces, as set up by Fefferman, Graham, Witten, Yau, Anderson and others, and go on to discuss recent work concerning existence, regularity and uniqueness questions, behaviour of renormalized volume as a function on the moduli space and properties of the Anderson degree.

March 14, 2003
Russel Caflisch, U.C.L.A.
Dynamics of a Step Edge in Thin Film Growth
Epitaxial thin films grow by attachment of adatoms to step edges (or island boundaries). In contrast to the assumptions of classical models, the state of a step edge is typically in a kinetic steady state that is far from equilibrium. This talk presents a detailed model for the dynamics of a step edge, along with analysis of the model in several limits, and a discussion of equilibrium for this system. The model is partially validated by comparison to results from kinetic Monte Carlo simulations. For large adatom diffusion, the asymptotics of this model includes edge diffusion and line tension, which provides an atomistic, kinetic derivation of the Gibbs-Thomson formula.

April 11, 2003
Peter Constantin, University of Chicago
Remarks on Rotating Fluids
I will give a nonlinear version of the Taylor-Proudman theorem concerning rotating incompressible Euler equations. I will discuss also the energy spectrum in the inverse cascade regime, for quasi-geostrophic equations.

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