Interactions Of Combinatorics – Icomb

The goal of this proposal is to create a strong team at the Université Paris-Sud
to study the interactions between combinatorics and other fields, such as number theory,
basic hypergeometric series, representation theory, geometry and mathematical physics.
We will study these interactions from the perspective of two combinatorial objects that
are generalizations of integer partitions. The theory of integer partitions is a thrilling
field that has been developed primarily in the USA by combinatorialists, algebraists,
analysts, and number-theoreticians such as G.E. Andrews, B. Berndt, K. Ono, R. Stanley,
J. Lepowsky, their students and many others. A number of strong young researchers in
partition theory have recently relocated or returned to Europe. More than ever, the time
is now right to begin building a strong European group. For the first time, in 2006 in
France, a workshop focusing on partitions in Europe was organized, with 10 speakers and
some 20 participants. Over the coming 5 years we shall demonstrate the success of our
project by organizing the next editions of this workshop and building it into a regular
conference of international stature, reflecting the emerging role of European researchers
as leaders in the field.
The two aforementioned generalizations of partitions that will form the focus of our
investigations are permutation tableaux and overpartitions. Permutation tableaux arise
from the enumeration of totally positive Grassmann cells in algebraic geometry and they
have been recently found to play a key role in certain statistical physics models. Overpartitions
come from the combinatorics of basic hypergeometric series and have many
connections to number theory, algebra and mathematical physics. We believe that overpartitions
are also the most natural objects to study the representation theory of Lie
superalgebras. Over the next 5 years we shall demonstrate that much of the classical
theory of partitions is part of a broader picture involving these new objects.
Our present team is composed of 5 members (2 tenured researchers, 1 postdoctoral
researcher and 2 PhD students) as well as 6 international collaborators who are assistant
and associate professors (from the Netherlands, Ireland, Korea and USA). We are all
thirty-five years old or younger and half of the team is comprised of women. Together we
have been responsible for over 100 scholarly publications.