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ZiF Research Group

Stochastic Modeling in the Sciences

Stochastic Partial Differential Equations and Random Media


Convenors: Friedrich Götze, Yuri Kondratiev, Michael Röckner (Bielefeld)

The research semester started April 4 - 8, 2005 with the opening conference Stochastic partial differential equations and random media: From theory to applications I and ended September 12 - 16, with the second part, the concluding conference. In between these two main conferences five further workshops took place. Altogether, in the course of the research semester, approximatively 140 internationally acknowledged scientists came to Bielefeld to attend the conferences, and among them about 25 researchers took the opportunity of a longer stay at the ZiF and participated as so-called "fellows in residence" in the research group.
The research semester was the central activity of the research group. The focus of this group lies on the interaction/cooperation between theory and applications in two important fields of probability theory, in fact two reasearch directions which have attracted more and more attention in the recent years-the theory of stochastic (partial) differential equations and the theory of random media.
On the mathematical side the respective theories are becoming more and more established. On the applied side the demand for good mathematical theories to solve real-life-problems is growing. But the development of a common language to render possible the exchange and communication between the different communities, however, is not at grade. The research group took into account this demand of development, bringing together scientists using stochastic modelling, based on stochastic partial differential equations and/or random media, in their research on the one side, and internationally leading probabilists working in those two fields on the other.
This intention is also reflected by the huge range of topics treated during the seven conferences and workshops. These varied from physics, quantum chaos and random media to climatology, economics, biology and chemistry, where the theoretical as well as the applied side was highlighted.
During the opening conference the invited speakers gave an overview over the newest trends, concepts and developments, theoretical results and applications on the fields of SPDE's and random media. Concerning the applications, the emphazis was laid on those which would play a major role for the research activities and following workshops, i.e., as already mentioned above, applications from physics, quantum chaos, climatology, economics, biology and chemistry. At the beginning the research activities of the fellows concentrated around random energy, spin glass models and random polymers. For example, Francesco Guerra (Rome 1) gave an elaborate lecture series about astonishing progress that has been made towards the mathematical understanding of structural aspects of mean field spin glass theories. He presented results, most recently achieved in this area. His lecture series provided a basis and a profound preparation for the following workshop on Spin glasses and scaling limits which was held from May, 2 - 4, 2005.
This workshop was followed May 23 - 25, 2005 by a workshop about Quantum chaos, random matrices and their applications. Impressive results were presented. For instance J. Marklof (Bristol) explained that eigenfunctions of the Laplacian on certain non-compact domains with finite area may be localized at infinity, provided there is no extreme level clustering, and thus rule out quantum unique ergodicity for such systems. His construction is elementary and based on "bouncing ball" quasimodes whose discrepancy is proved to be significantly smaller than the mean level spacing.
Subsequently, a threepart workshop series on applications of the theory of SPDE's and random media to real-world-phenomena as can be found in climatology, economics and biology, took part. In particular, the group around P. Tass from the research center in Jülich joined the research group. The topics of these series are so extensive that they can not be summarized briefly at this place. The interested reader may visit the homepage of the research group to get an overview.

The research semester was closed by the concluding conference Stochastic partial differential equations and random media: From theory to applications II, where a highly animated round table discussion about stochastic dynamics of interacting particle systems pointed to new models, problems, and methods in the area.
Apart from the conferences and workshops, the work of the research group consisted in weekly meetings for seminars and round tables. These additional meetings took place each Wednesday afternoon. Here, specially invited speakers, fellows and even young local scientists reported on their research and latest results. During the summer months, the weekly seminar had to be expanded by another supplementary day, to take into account the high number of fellows in residence, who were interested in a scientific exchange.
To facilitate this exchange, the outcomes of the workshops, conferences and seminars were documented. On the webpage of the ZiF an essential number of preprints reflecting the current results is available.

Requests on the Research Group Stochastic Modeling in the Sciences was answered by the scientific assistant Dr. Gerald Trutnau