Past Colloquia

The recent developments of Mathematical Physics have brought very new ideas about the nature of space . I will argue that we have to mix the methods of noncommutative geometry of Alain Connes with the prophetic views of Grothendieck about the so-called motives and their motivic Galois group .
The dream of a "cosmic Galois group" may soon become an established reality .
 

I will give an expository account of Mathai, Melrose, and Singer [math.DG/0402329], explaining how to define the projective Dirac "operator" when the underlying manifold is neither spin nor spin_C, and how to define its analytic index which need not be an integer. Nevertheless, the usual index formulas apply. Professor Singer will be admitted as honorary member of the London Mathematical Society just before his talk.

Most people acquire their

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.

Why does a spinning coin come to such a sudden stop? Why does a
spinning hard-boiled egg stand up on its end? And why does the
rattleback rotate happily in one direction but not in the other?
The key mathematical aspects of these familiar dynamical phenomena,
which admit simple table-top demonstration, will be revealed.