More than 100 years after its birth, representation theory is a vibrant, growing subject, and
a way of thinking that has been valuable in many very different areas of mathematics. It is a very broad topic which crosses many fields including Lie algebras, finite and algebraic groups, combinatorics, geometry, integrable systems, analysis, category theory, knot theory, mathematical physics and number theory.
The proposed Durham symposium will concentrate on combinatorial, algebraic and geometric aspects of representation theory, and will bring together mathematicians working in representation theory from these directions, with an aim of finding common ground and discussing exciting recent developments.
The symposium will bring together leading
for a 10 day period of concentration,
consolidation and cross-fertilization.
The programme will be organised to focus on recent breakthroughs and methods in representation theory, particularly where interactions between different specialities are likely to be beneficial.
The topics have been chosen to reflect areas where there has been
significant recent progress and where there is much high-level
international activity: semisimple Lie algebras, quantum groups and crystal bases; algebraic groups and Lie algebras in positive characteristic; Hecke algebras; quivers; cluster algebras; group representation theory. The headings are broad and there is a
great deal of overlap between them. For instance the crystal base
theory for Fock space representations of (quantum) affine
algebras, a combinatorial area of Lie theory, plays a significant
role in all these sections; the LLT conjecture on decomposition
matrices of Hecke algebras, which was proved by Ariki, was motivated by this. There are deep connections to
symmetric function theory which is another of the threads tying
the proposal together; symmetric functions have been at the centre
of representation theory since the first study of the
representation theory of symmetric groups by Frobenius and Schur.
Here Schur polynomials and their generalisations to Macdonald
polynomials provide many research directions: finding
combinatorial descriptions of representation theoretic data; using
representation theory and/or geometry to prove positivity claims,
etc. There is great interest in understanding combinatorial
descriptions of these polynomials, and of generalising them and
the corresponding background theory from symmetric groups to other
Weyl groups. This crosses all the topics above and unifies much of
representation theory and combinatorics.