FOR 498:  Dutch-German Bilateral Research Group on: Mathematics of Random Spatial Models from Physics and Biology

• Subject Area: Mathematics
• Term: from 2003 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5470024

The goal is to develop the mathematics of random spatial models from physics and biology. The research focusses on three key topics that are closely related:-- random media;-- randomness in population evolution;-- random matrices.Random media:Many physical and biological systems are so complex on a microscopic scale that the only efficient way to obtain some insight into their macroscopic behaviour is by modelling them as systems with random interactions. The problems encountered in their analysis, both analytically and numerically, prove to be a tremendous challenge for mathematics and to provide considerable feedback for the advancement of modern probability theory.Randomness in population evolution: In the theory of stochastic population models, recent progress has been made in studying the effect of branching and/or resampling in combination with migration, mutation, selection and recombination, both for Moran models and for Fleming-Viot models. Further recent progress concerns spatial critical branching systems where the branching rate is not constant but depends on the type of the individual and on the local state of the system, the latter effect being due to finite resources and/or competition. The long-time behaviour of such systems is ill understood, in particular, the multiscale space-time features. However, new methods have emerged that seem promising and are developed further. Different regimes are of interest. In one regime there are strong self-regulatory forces, allowing for locally small populations, in another regime there is reinforcement, allowing for locally large populations. The main challenge in this context is to better understand the role of universality in relation to the "genealogical aspects" of a population, which determine the type composition of the population.Random matrices:In recent years much effort has been devoted to the study of the spectrum of large random matrices. The conjecture is that for many examples the spectrum falls into one of a limited number of so-called universality classes. So far this universality has been demonstrated on only a few examples. Key words are the Wigner semi-circle distribution and the Tracy-Widom distribution. Applications are widespread (e.g. in high-energy physics, statistical data, population dynamics). Random matrices also come up in number theory, random tilings, and (de)coding of random words, thus providing a deep link at an algebraic level.

• DFG Programme: Research Units

Projects

• 1. Random matrices 2. Stochastic networks, and statistical recognition in biology (Applicant Baake, Ellen )
• Equilibrium and ageing in glassy systems (Applicant Bovier, Anton )
• Populations with self-regulated production and reproduction (Applicant Wakolbinger, Anton )
• Spatial multitype population models with type interaction (Applicant Greven, Andreas )

• Spokesperson: Professor Dr. Andreas Greven