http://math.northwestern.edu/Pinsky-poster-09.pdf

Abstract:
The analysis of the usual method of shuffling
cards leads to surprisingly neat results. It
also has connections with group theory,
combinatorics, and geometry. These
expository lectures are aimed at a general
mathematical audience and are largely self
contained and independent of each other.

I: Basic Shuffling
I will show that it takes seven ordinary riffle
shuffles to adequately mix up 52 cards.
The mathematics behind this is essentially
symmetric function theory and leads to things
like Hodge decompositions of Hochschild
homology. A variant to type-B shuffles is used
to break casino card shuffling machines.

II: Hyperplane Arrangements
The combinatorics of hyperplanes in
Euclidean space has a simple random walk
interpretation. This generates all kinds
of shuffling schemes (Bidigare, Hanlon,
Rockmore). Generalizations to walks on
the chambers of a building and semigroups
(Brown) illuminate calculations originating in
library science.

III: Adding Up a List of Numbers
The usual process of carries when adding
up a list of numbers can be analyzed using
an amazing matrix (Holte). This matrix has a
card shuffling interpretation. This leads to
new results about addition, shuffling, and the
Hilbert series of Veronese subrings. This is
joint work with Jason Fulman.