Santosh Vempala ( vempala@math.mit.edu )

Random walks have intrigued us for a long time. In recent years,
they have also played a crucial role in the discovery of
polynomial-time algorithms.  The talk will illustrate this using
random walks in Euclidean space, showing how they lead to a
simple algorithm for the fundamental problem of convex optimization.
Along the way, we will describe some engaging ways to walk randomly
and highlight their geometry.

From Descents to Peaks: An Alpine View of Enumeration
in Hyperplane Arrangements

Louis J. Billera, Cornell University
 

We diagonalize a map, defined by Stembridge, that associates descents to peaks
in the context of quasisymmetric functions. When restricted to peaks, it can be
viewed as giving a random walk on the collection of peak sets. The stationary
distribution of this walk is the distribution of peak sets over the symmetric group.
When applied to geometric lattices (matroids), it gives the complete enumeration of
chains of faces in any hyperplane arrangement having that lattice as its lattice of
intersections, extending the classical results of Zaslavsky.

Towards pictures of Kazhdan-Lusztig polynomials

Gregory Warrington (UMASS, Amherst)
 

Kazhdan-Lusztig polynomials encode detailed information about both
representation theory and the geometry of Schubert varieties. In the
most important cases, it is known that their coefficients are
non-negative integers. However, a longstanding problem is to find a
_combinatorial_ description of these coefficients. In this talk we
give such an interpretation for the coefficient of the linear term in
any Kazhdan-Lusztig polynomial associated to two permutations. Our
description relies on simple pictures that can naturally be associated
to any pair of permutations.

 Dan Willard    (University at Albany, SUNY)

Godel's Second Incompleteness Theorem states that sufficiently strong
axiom systems are unable to prove a formal theorem asserting their own
consistency.  This talk will discuss some new generalizations and
exceptions to the Second Incompleteness Theorem that we have recently
developed and published during the last two years in the Journal of
Symbolic Logic. Our talk will explain that there are certain senses in
which an axiom system of moderate strength can verify its own consistency.