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.