Professor Xavier Viennot from LaBRI, University of Bordeaux will offer a twelve hour mini-course on Algebraic Combinatorics from January 12–29, 2010. All lectures will be held in the Seminar Hall at CMI.
List of topics
A spectacular renaissance is happening in combinatorial mathematics. One of the main motivation is the resolution of purely enumerative problems, often motivated by questions coming from other domains such as theoretical physics, analysis of algorithms in computer science or molecular biology. The main tool is the notion of generating function. Recurrence formulae, functional or differential equations, rational and algebraic power series, etc, are everywhere in this domain called enumerative combinatorics. It will be the subject of the first two classes.
More recently, some theories appeared for a better unifying and understanding of certain formulae or some calculus from enumerative combinatorics. Some combinatorial models appeared and the domain began to be organized. The so-called bijective combinatorics play a central role and appears to be a new paradigm for looking and interpreting in a combinatorial way some part of classical mathematics. The interaction between combinatorics and algebra gave birth to algebraic combinatorics, a domain nowadays very active. The aim is to solve some combinatorial problem using algebraic technics, or conversely give some interpretations of algebraic theories using some finite combinatorial structures. The four others classes develop these considerations, in relation with theoretical physics and computer science.
This new combinatorics has a direct and fructuous relationship with theoretical physics. Some physicists call this new domain combinatorial physics or integrable combinatorics. It is related to statistical mechanics, with historically the Ising model in dimension 2, and for example more recently hard gas model, directed animals or Lorentzian quantum gravity. Very recently the famous Razumov-Stroganov conjecture relating quantum spins chains of XXZ Heinsenberg model with the combinatorics of alternating sign matrices and plane partitions, has attracted much attention in the theoretical physics community together with the combinatorists.