L2

Einstein bequeathed many things to differential geometry — a
global viewpoint and the urge to find new structures beyond Riemannian
geometry in particular. Nevertheless, his gravitational equations and
the role of the Ricci tensor remain the ones most closely associated
with his name and the subject of much current research. In the
Riemannian context they make contact in specific instances with a wide
range of mathematics both analytical and geometrical. The talk will
attempt to show how diverse parts of mathematics, past and present,
have contributed to solving the Einstein equations.

Stability is central to the study of numerical algorithms for solving
partial differential equations. But stability can be subtle and elusive. In
fact, for a number of important classes of PDE problems, no one has yet
succeeded in devising stable numerical methods. In developing our
understanding of stability and instability, a wide range of mathematical
ideas--with origins as diverse as functional analysis,differential geometry,
and algebraic topology--have been enlisted and developed. The talk will
explore the concept of stability of discretizations to PDE, its significance,
and recent advances in its understanding.

A little more than 100 years ago, Issai Schur published his pioneering PhD
thesis on the representations of the group of invertible complex n x n -
matrices. At the same time, Alfred Young introduced what later came to be
known as the Young tableau. The tableaux turned out to be an extremely useful
combinatorial tool (not only in representation theory). This talk will
explore a few of these appearances of the ubiquitous Young tableaux and also
discuss some more recent generalizations of the tableaux and the connection
with the geometry of the loop grassmannian.

Aggregation refers to the thermodynamically favoured coalescence of individual molecular units (monomers) into dense clusters. The formation of liquid drops in oversaturated vapour, or the precipitation of solids from liquid solutions, are 'everyday' examples. A more exotic example, the crystallization of hydrophobic proteins in lipid bilayers, comes from current biophysics.

This talk begins with the basic physics of the simplest classical model, in which clusters grow by absorbing or expelling monomers, and the free monomers are transported by diffusion. Next, comes the description of three successive 'eras' of the aggregation process: NUCLEATION is the initial creation of clusters whose sizes are sufficiently large that they most likely continue to grow, instead of dissolving back into monomers.

The essential physical idea is growth by unlikely fluctuations past a high free energy barrier. The GROWTH of the clusters after nucleation depletes the initial oversaturation of monomer. The free energy barrier against nucleation increases, effectively shutting off any further nucleation. Finally, the oversaturation is so depleted, that the largest clusters grow only by dissolution of the smallest. This final era is called COARSENING.

The initial rate of nucleation and the evolution of the cluster size distribution during coarsening are the subjects of classical, well known models. The 'new meat' of this talk is a 'global' model of aggregation that quantitates the nucleation era, and provides an effective initial condition for the evolution of the cluster size distribution during growth and coarsening. One by-product is the determination of explicit scales of time and cluster size for all three eras. In particular, if G_* is the initial free energy barrier against nucleation, then the characteristic time of the nucleation era is proportional to exp(2G_*/5k_bT), and the characteristic number of monomers in a cluster during the nucleation era is exp(3G_*/5k_bT). Finally, the 'global' model of aggregation informs the selection of the self similar cluster size distribution that characterizes 'mature' coarsening.