In this talk we present some results of our recent efforts (jointly with S.
Sehgal, I. Shestakov and M. Zaicev) to classify all possible abelian group
gradings of finite-dimensional simple algebras from various classes,
including those of associative, Lie and Jordan algebras. In a number of
cases this classification is essentially complete.

A classical result in SCV is the fact that a nonconstant holomorphic map sending a piece of the unit sphere in $\\ C^N$ into itself is necessarily locally biholomorphic (and, in fact, extends as an automorphism of the unit ball). Generalizations and variations of this result for mappings between real hypersurfaces have been obtained by a number of mathematicians over the last 30 years. In this talk, we shall discuss some recent joint work with L. Rothschild along these lines for mappings between CR manifolds of higher codimension.

Joint Seminar with UCI

The study of a tree introduced by Lascoux-Schutzenberger
yields a family of tableaux correspondences which can be
used to enumerate the number of reduced factorizations
of a permutation. Representing reduced words by certain line diagrams,
these corresponences can be obtained by an algorithm of D. Little
in a remarkably simple manner. The talk uses a beautiful Applet
also created by D. Little which illustrates the algorithm and
should be helful in any further studies of reduced words.

The ancient Greek mathematicians did not use limits. We will
examine their proofs of some propositions for which our proofs would
involve limits. The discussion will show that their proofs helped the
reader gain insight neither into how these theorems were discovered nor
into how new results might be found. Some historians thought that calculus
was being used to make the discoveries, even though it was not used in
their proofs. Indeed, in about 1906, a copy of a book of Archimedes\'s
book, \"The Method\"" was discovered. Archimedes revealed his process of

This is a talk on heuristic aspects of Idempotent Mathematics
in the spirit of current works of V.P. Maslov and his collaborators.
Idempotent Mathematics can be treated as a result of a dequantizati-
on of the traditional Mathematics as the Planck constant tends to
zero taking pure imaginary values. For example, the field of real
numbers can be treated as a quantum object whereas idempotent semi-
rings can be examined as \"classical\"" or \""semiclassical\"" objects

I will first recall some examples of L-functions and indicate some of the
ways they have been important in algebraic number theory. I will then
describe what appears to be their intimate connection with Galois theory
(eg the Fontaine-Mazur conjectures), as well as touching on their
relationship with algebraic geometry and automorphic forms. Finally, I will
discuss what can be proved in this direction.