Recent Advances in Algebraic and Enumerative Combinatorics

Algebraic and enumerative combinatorics is concerned with objects that have both a combinatorial and an algebraic interpretation. It is a highly active area of the mathematical sciences, with many connections and applications to other areas, including algebraic geometry, representation theory, topology, mathematical physics and statistical mechanics. Enumerative questions arise in the mathematical sciences for a variety of reasons. For example, non-combinatorial structures often have a discrete substructure when they are studied up to topological equivalence. Moreover, some non-combinatorial structures can be discretized, and are therefore susceptible to combinatorial techniques, and the original question is then recovered by the appropriate limit taking (the dimer problem in statistical mechanics is an instance of this). Algebraic and enumerative combinatorics is a highly active area of the mathematical sciences, and was the subject of a special year at MSRI in 1996/97, organized by Billera, Bjorner, Greene, Stanley and Simion. The one week workshop at BIRS focussed on three topics, discussed in detail below, in which substantial progress has been made in the last five years. The topics are interrelated, as is essential for a workshop of this length. The workshop was successful in bringing together a very strong international collection of active researchers in algebraic and enumerative combinatorics, as well as other areas of the mathematical sciences in which substantial enumerative questions with a strong algebraic or analytic foundation have arisen. The connections that have been made between algebraic combinatorics and other areas in the last few years have contributed to rich and significant lines of research. Such research requires a familiarity of several research disciplines, and the workshop facilitated this contact between algebraic and enumerative combinatorics and other fields in an effective way.