Christian Krattenthaler,
Universität Wien:
Enumeration of Plane Partitions and Alternating Sign Matrices
Plane partitions and alternating sign matrices have been among the most intensively studied objects in enumerative combinatorics over the past 40 years. In particular, the programme of enumeration of symmetry classes of plane partitions and alternating sign matrices has fascinated researchers, since the proofs of the (initially) conjectured enumeration formulae posed significant challenges, of both methodological and technical nature. Only recently has this programme been completed.
I shall start by presenting these two classes of combinatorial objects, together with an outline of the most important methods to enumerate them. Although it may seem on the superficial level that "all problems have been solved", this is not at all so, as I will explain in the third part of these lectures
Stephanie van Willigenburg, University of British Columbia:
Quasisymmetric Schur functions
(I, II, III, IV)
Thursday 8th-Friday 9th: 11:30-12:30 and 16:30-17:30, Sala 2.4.
In algebraic combinatorics a central area of study is Schur functions. These functions were introduced early in the last century with respect to representation theory, and since then have become important in other areas such as quantum physics and algebraic geometry.
These functions also form a basis for the algebra of symmetric functions, which in turn forms a subalgebra of the algebra of quasisymmetric functions that itself impacts areas from category theory to card shuffling. Despite this strong connection, the existence of a natural quasisymmetric refinement of Schur functions was considered unlikely for many years.
In this short course we will meet such a natural refinement of Schur functions, called quasisymmetric Schur functions. Furthermore, we will see how these quasisymmetric Schur functions refine many well-known Schur function properties, with combinatorics that strongly reflects the classical case such as diagrams.
This course will require no prior knowledge of any of the above terms.
