Interacting stochastic systems consist of a large number of interacting
random components. These components interact with each other and with
their environment. Even when the interaction is local, such systems
typically exhibit a complex global behavior, with a long-range
dependence resulting in anomalous fluctuations and phase transitions.
To mathematically understand these systems requires the use of powerful
probabilistic ideas and techniques. The challenge is to introduce
simple models, which serve as paradigms, and to unravel the complex
random spatial structures arising in these models. Statistical physics
and ergodic theory provide the conceptual ideas, while probability
theory provides the mathematical language and framework. The important
challenge is to give a precise mathematical treatment of the physics
that arises from the underlying complexity. Much of the knowledge that
has been built up in mathematical statistical physics over the past
decades is currently making its way into biology. One of the tasks is
to help facilitate this cross-fertilization and to address concrete
biological questions at the interface. Examples are coming from
population genetics and immune system biology.
The research in Probability Theory concentrated on interacting
stochastic systems (disordered systems, percolation, random polymers,
metastability, sandpiles), ergodic properties of random processes
(dynamical Gibbs-non-Gibbs transitions, hidden Markov chains), and
topics from mathematical biology (population dynamics, T-cells). Key
tools are large deviation theory, stochastic analysis, variational
calculus and combinatorics. There is an interesting link between
algebraic dynamical systems and solvable models of statistical
mechanics. It turns out that entropies of apparently different systems
often coincide, and that this `mere' coincidence is not accidental.
Research aims at providing an explanation for this phenomenon. A
powerful combinatorial technique to study high-dimensional systems is
the 'lace expansion'. Research aims at obtaining a rigorous
understanding of phase transitions in high dimensions, including
diffusion on critical spatial structures.
The research in Operations Research concentrated on Markov chains,
Markov decision processes and Markov games, with applications to
problems in stochastic networks. One of the main issues concerns
stability. How can stability be checked? If stable, how fast does the
network reach its stationary distribution? If unstable, what does the
quasi-stationary distribution look like? How can efficient algorithms
be developed to control the network according to certain pre-set
optimization criteria? Are these algorithms amenable to practical
implementation? What can one say about the structure of optimal
policies? Which type of customer should be prioritised to optimise
network performance? These questions can be studied within the
framework of Markov chain theory.
Often the situation arises where there are conflicting interests, for
instance, maximizing server efficiency while minimizing customer
dissatisfaction. This may be studied through Markov game models.