In the representation theoretic interpretation of the theory of
automorphic forms Fourier transforms at cusps are products of two
quantities. The first (under a multiplicity one condition) is a scalar
containing all of the arithmetic information. The second is a (generalized)
Whittaker model for the representation associated with the form. In this
lecture we will analyze the integrals involved in the second part of this
factorization. These integrals are paramaetrized by points in a complex
vector space and converge and are holomorphic in a half space. The main
result gives an algebraic condition that guarantees a holomorphic
continuation to the entire space. This result generalizes or implies every
known case of a holomorphic continuation of a generalized Jacquet integral.

An innovative connection between symmetric functions and the study of
permutation enumeration will be described. Generating functions will be
produced which enumerate permutations (and other Coxeter groups) by natural statistics. These techniques consolidate many classic results and give new information about interesting subsets of the symmetric group such as 321-avoiding and alternating permutations. This is the first of two talks
on the subject.

A combinatorial game is a perfect information game with no chance played
by two players who take turns making moves -- the last player to move wins
the game. There is an algebraic system associated with combinatorial game
theory that features bizarre objects such as infinitesimals -- things that
are positive, yet so small that any sum of them, no matter how many, is
not bigger than any positive number. The real numbers do not have
infinitesimals, but combinatorial game theory is rife with them. While
their structure can be baffling at times, the ideas are very simple.
We\'ll play some games -- and very TINY ones at that! You need to know
nothing to understand the majority of this talk -- we\'ll introduce
everything we need during the talk.

Refreshments will be served.

In a short paper in 1861 Hermite showed that a general equation of degree 5,
$$
x^5 + a_1x^4 + a_2 x^3 + a_3x^2 + a_4x + a_5=0,
$$
can be reduced to the form $x^5 + ax^3 + bx + b = 0$. Since then, this
result and related questions have been studied from different viewpoints,
by Felix Klein, David Hilbert, Richard Brauer, Jean-Pierre Serre, Yu I.
Manin and others. More recently, Joe Buhler and Zinovy Reichstein found a
very interesting connection of these problems with the study of rational
covariants of the symmetric group.
We will explain this approach and show how it is related to some classical
invariant theory. (This is joint work with G.W. Schwarz.)