Nonlinear Diffusions and Entropy Dissipation: From Geometry to Biology (10w5054)

Objectives

The proposed workshop is in part a follow-up to the workshop "Nonlinear diffusions: entropies, asymptotic behavior and applications" held at Banff in April 2006. It is motivated by the rapid progress in the field and the emergence of new applications and connections between seemingly unrelated areas. The intent is to bring together researchers working in various fields, connected by the common mathematical structure of the models, and a core group of mathematical experts in entropy based methods. Fruitful exchanges between experts in entropy methods at theoretical level, and more applied scientists dealing with models that applications have set forth, are expected.

The array of nonlinear systems with diffusive behavior is enormous. The geometric viewpoint based on entropy and its dissipation provides a unifying framework which enables for well-established techniques to be implemented in investigating active and emerging applications such as nonlocal diffusive systems in biology, phase transitions models, equations for modeling the gravitational interaction of particle clouds, and mean-field games.

The variety of dissipation mechanisms generates a variety of geometries in their mathematical descriptions. Applications in biology, image processing (morphometrics), thin liquid films, pattern evolution and fluid mechanics highlight the need for a better understanding of the geometry of the energy landscape for dissipation mechanisms that do not fall in the presently well studied cases of Sobolev spaces and the Euclidean Wasserstein space.

On the other end of the spectrum are the applications to differential geometry and to diffusive equations driven by the geometry of the underlying manifold. The rapid progress on curvature driven flows set off by the works of Perelman together with strong connections with nonlinear diffusions and recent progress on studies of fast-diffusion equations present a real opportunity for interaction.

Although aspects of nonlinear diffusions have been treated at conferences devoted to PDEs, kinetic theory, geometrical flows, or mathematical biology, no workshops, apart from the one in Banff in 2006, focused on the transversal relation between these fields with common core: nonlinear diffusions and entropy methods. We believe that such an interaction is important and genuinely contributes to each community. The workshop will provide an opportunity for young researchers to get a broad overview of the field and substantially interact with other researchers. The number of talks will be limited to provide plenty of time for scientific discussion.