| The rigorous mathematical study of Random Spatial Processes (in particular percolation and interacting particle systems) has become one of the most active subfields in modern probability. Three semester-long special programmes will be held in 2008 and 2009 with this area of research as one of the main topics: at the Isaac Newton Institute (Cambridge), the Institut H. Poincar\'e (Paris) and the Mittag-Leffler Institute (Stockholm). Motivation and inspiration comes from problems in the Earth and Life Sciences(e.g. population dynamics, aspects of neuron activity,forest fires), Physics (rigorous description of phase transitions) and, more recently, Computer/Communication Technology (connectivity of ad hoc networks). To get an impression of the flavour, mathematical techniques and current state of this area, see e.g. \cite{Li99}, \cite{Gr99}, \cite{BoRi06c}, and \cite{We04}. Inequalities have always played an important role in this field, for instance to show certain asymptotic identities by suitable bounds for `error terms'. A classical example is the FKG inequality, which is used `all over the place'. A more recent example, involving techniques from several parts of mathematics, is the collection of inequalities that has been (and is being) developed to explain and quantify so-called sharp-threshold phenomena that occur in many proccesses. This proposal, involving a PhD student, presents a variety of scientifically challenging problems where such or other inequalities (are expected to) be important. It will depend on the background and interest of the PhD student, and on new developments, which of these problems will be selected |
