Courses:

(Click on the title for further information about the course)

The combinatorics of polyhedral divisors. Klaus Altmann (Freie Universität Berlin, Germany)

The theory of so-called toric varieties is a well established tool to study algebro geometric objects by combinatorial methods (and vice versa). It translates varieties with a (full) torus action into cones, polyhedra, and fans. Moreover, many of the properties of the varieties have a combinatorial counterpart via this correspondence.
 
A full torus action means that the torus has the same dimension as the variety in question. However, in some situations, varieties with a lower-dimensional torus action arise in a natural way: The striking example is the total spaces of deformations of toric varieties. For those situations we show how to adapt the toric language to keep a description involving as much combinatorics possible. The associated notion is that of a combinatorial divisor developed together with Juergen Hausen.

Primary decomposition of binomial ideals. Ezra Miller (Duke University, USA)

Primary decomposition of ideals generated by binomials comes with inherent lattice-point combinatorics. The central idea is to construct graphs out of the set of monomials, using binomials as edges. The connected components of the graphs associated to a binomial ideal control its primary decomposition. Polyhedral combinatorics permeates the theory, most clearly when the ideal is prime: the connected components comprise the lattice points in a convex polyhedron. Potential applications of binomial ideals to be discussed could include hypergeometric series, or combinatorial games, or dynamics of chemical reactions under mass action kinetics.

Symmetric function approach to representation theory and applications.. Mercedes H. Rosas (Universidad de Sevilla, Spain)

This course will deal with the following topics:
- Schur's thesis. The representation theory of the symmetric group. The language of symmetric functions.
- Schur-Weyl duality. The representation theory of the general lineal group and related groups.
- The Littlewood-Richardson coefficients. The saturation theorem of Knutson and Tao. The Kronecker coefficients.
- Murnaghan's reduced Kronecker coefficients.
- Quantum information theory : applications to the problem of entanglement.
During the course of the lectures there will be time allocated for problem sections.