14 January - 4 July 2008

**Organisers:** Professor PJ Cameron *(Queen Mary, London)*, Professor B Jackson *(Queen Mary, London)*, Professor A Scott *(Oxford)*, Professor A Sokal *(NYU/UCL)* and Professor DG Wagner *(Waterloo)*

The past half-decade has seen an increasing interaction
between combinatorialists, probabilists,
computer scientists and theoretical physicists concerned broadly
with the study of "probability theory on graphs"
or "statistical mechanics on graphs".

The programme will build on this cross-fertilisation. It is particularly timely for a number of reasons:

- methods from mathematical physics are beginning to make their mark on previously intractable combinatorial problems;
- increasing computer power, together with the wide availability of symbolic-algebra packages, has brought the possibility of exploration of non-trivial examples;
- phase transitions are increasingly being investigated on a wide variety of combinatorial structures, including matroids, set partitions and constraint satisfaction problems, as well as graphs.

In this interdisciplinary field, the questions being investigated typically arise naturally in combinatorics or probability, but key elements of the intuition needed for their solution often come from physics, for example from the theory of phase transitions and critical phenomena.

Particular topics to be addressed by the programme and its workshops will include: zeros of combinatorial polynomials, including the chromatic, flow, reliability and Tutte polynomials; Markov-chain Monte Carlo methods; combinatorial identities and their applications in statistical mechanics; use of methods from statistical mechanics and quantum field theory in combinatorial enumeration; correlation inequalities; and phase transitions in combinatorial structures.

Limited travel and subsistence funds are available from an NSF grant to support the participation of researchers (of any nationality) affiliated to a U.S. institution. The aim is to facilitate the participation of those who may not have access to alternative funding sources for their travel and subsistence, particularly young researchers, women, minorities, and researchers based at small colleges.

If you wish to be considered for an NSF grant, please complete the attached form

Applicants for support should include a brief letter indicating their mathematical interests and reasons for wishing to participate in the programme, their first and second choices for dates of stay, and their other funding sources (if any). They should also send a CV and, in the case of graduate students, a letter of recommendation from the dissertation advisor.