Warwick EPSRC Symposium in the Statistical Mechanics / Mathematics of Phase Transitions

This is a proposal for a symposium to be held during the academic year 2013-2014 at the Mathematics Research Centre (MRC) of the University of Warwick. The overall theme is mathematical statistical physics with a particular emphasis on the extension of its methods to a range of disciplines within mathematics as well as its applications in physics and other sciences.

Mathematical statistical mechanics developed from its focus on rigorous foundations of statistical physics to become an exciting and active area of research devoted to various aspects of collective phenomena. Remarkably, its methods designed originally for exploring phase transitions are providing effective tools in other branches of mathematics: combinatorics, discrete mathematics, probability, analysis, and computer science. They turned out to be especially useful for investigations of asymptotics with abrupt thresholds and universal behaviour at criticality, the topics brought together under the heading "Mathematics of Phase Transitions". This is exemplified by impressive successes in various disciplines.

a) In probability: studies of universal critical behaviour of various two-dimensional systems based on Schramm's SLE led to spectacular advances
and opened a whole new field of modern mathematics with Fields Medals awarded in 2006 and 2010.

b) In discrete mathematics and computer science: the theory of random discrete structures that is crucial for these disciplines has been truly
revolutionised by new ideas from statistical mechanics.

c) In analysis: investigations of macroscopic nonlinear PDE, as applied, for example, in nonlinear elasticity or in investigations of phase
coexistence, based on underlying microscopic atomic models are gaining prominence.

The symposium will facilitate a year-long sustained research activity on mathematical aspects of statistical physics with emphasis on its links to other mathematical disciplines. It will foster international links and collaborations and attract the attention of the UK mathematical community to this newly emerging and highly active area and enable the innovative use of the potential of branches of mathematics that are well established in the UK.

In practical terms the symposium will be structured around six week-long workshops in various research areas of international significance pertaining to the general theme of the symposium. All important areas of mathematics where the methods of statistical physics play a significant role will be covered, with participation of leading international and national experts.
In addition, there will be a visitor programme and a number of other supporting activities (mini-courses, seminars) designed to sustain the level of activity through the year.

Statistical Mechanics and phase transitions play a fundamental part in our understanding of the world and have had proven impact on every branch of science, engineering and social sciences. Methods of statistical mechanics are used everywhere: material science, computation, the weather and climate change, demographics, economics, communications, cell biology, particle physics, etc. However, it is also a mathematical discipline in its own right with deep connections across the spectrum of pure and applied mathematics and relevance in probability theory (large deviation theory, critical phenomena, many-body systems), discrete mathematics and computer science (message passing algorithms, low density parity check codes), analysis (variational problems, PDEs, quantum mechanics). This symposium will bolster the scientific base of the subject and ensure continued impact.

The main impact of national importance of this symposium will be on the mathematical community by

- Strengthening the multidisciplinary knowledge base and links within the concerned branches of mathematics in the UK;
- Bringing the best people worldwide to the UK in order to enhance its research basis and to promote their collaboration with UK researchers (including
postdocs and post-graduate students);
- Maintaining UK leadership at the cutting edge of research in this active and vibrant area.

In particular, this project will support UK universities drive to be among the prime generators of top-quality research. Of course this is not only a matter of prestige, as no country in the world can expect to maintain long term economic success without a strong scientific base and strong universities and, in particular, without a strong mathematical foundation.

Although the primary beneficiaries of this project will be researchers working in mathematics in general, and mathematical statistical mechanics and applied mathematics in particular, we envisage that the symposium will affect some areas of combinatorics, material science, economics, biology, physics, computer science and engineering. Several workshops will indeed explore the connections to these disciplines. In all these areas of research, the threshold phenomena like phase transitions play a central role and can be studied and analysed with the methods and basic principles of mathematical statistical mechanics.

In computer science, an area where the mathematics of phase transitions to be developed in the framework of the Symposium has a potentially immediate impact is the development of low density parity check codes (LDPC). This is one of the rare success stories where mathematical progress has quite directly had a significant practical commercially successful impact. The second area where this project has a potential practical impact is efficient Satisfiability Problems solving (SAT solving). Message passing algorithms based on statistical mechanics ideas seem to outperform standard methods on various important benchmarks. However, the theoretical understanding of message passing algorithms is currently very limited. To materialise the potential impact, we are going to proactively promote the symposium to the LDPC and SAT solving communities.