Department of Mathematics University of Coimbra Coimbra, Portugal
Persi Diaconis (Stanford University)

While representation theory is often developed as cutting edge pure
mathematics, it also has many applications. Some of these
are to other areas of mathematics (EG Group Theory or number
theory). In these lectures I will venture further a field,
to applied probability and statistics.
Lecture 1: The Mathematics of Shuffling Cards. How many
random transpositions does it take to mix up 52 cards? I
will show how character theory gives sharp answers to such
questions.
Lecture 2: Applications. As a follow up to the first talk, I will
show how the results apply to virtually any generating set
(comparison theory) and then to problems in statistical
physics (BernoulliLaplace urn) biology (phylogenetic trees)
and chemistry (coagulationfragmentation).
One of the most beautiful confluences of combinatorics and
representation theory is the theory of crystals:
combinatorial models for the internal structure of
representations of semisimpleLie algebras and Lie groups.
In the first talk I will introduce crystals
combinatoriallyno backgroundis needed for this, only a
willingness to watch me draw pictures. In the second talk
I will explain why the crystals are so powerful for
representation theory. In other words, the first talk will
be an introduction to the Littelmann path model, the
tableaux model and the LenartPostnikov model, and the
second talk will focus on applications: the Weyl character
formula, the Demazure character formula, the
LittlewoodRichardson rule, and the Ktheory of the flag
variety.