Entropies, the Geometry of Nonlinear Flows, and their Applications (18w5069)

Arriving in Banff, Alberta Sunday, April 8 and departing Friday April 13, 2018

Organizers

(Imperial College London)

(Rutgers University)

(Université Paris-Dauphine)

(TU-Munich)

(Carnegie Mellon University)

Objectives

The proposed workshop is in part intended as a follow-up to the workshops ``Nonlinear diffusions: entropies, asymptotic behavior and applications'', ``Nonlinear Diffusions and Entropy Dissipation: From Geometry to Biology" and ``Entropy Methods, PDEs, Functional Inequalities, and Applications", held at BIRS in April 2006, in May 2010 and in June 2014, respectively. Ideas emanating from stimulating exchanges at these events have led to important advances in optimality issues in the theory of functional inequalities, in problems of mathematical biology and geometric flows, with seemingly far-apart applications such as quantum Markov semigroups and collective behavior models. All in all there is an outstanding track record of this series of workshops as effective catalyzers for far-reaching consequences outside the comfort zone of mathematical applications of these tools.



This proposal is motivated by the fast-paced advance in the field and in particular by the need of examining new lines of research. We will proceed with the successful scheme of putting together different researchers of the ``core communities'' involved: PDE analysts, differential geometers and probabilists; and potential users of these techniques in pure and applied mathematics. Here are some emerging topics in entropy methods in PDEs and the study of their associated flows which will probably play an important role in the development of the field in the near future: - Reaction-Diffusion equations & systems as gradient flows. - Advances on gradient flows of non-geodesically convex functionals. - Symmetries of critical points not necessarily global minimizers of entropies. - Stability inequalities for remainder terms involving entropies and rates of decay. - Hypocoercivity & multiplicity of stationary states. - Discrete Entropies: Markov chains & clustering in machine learning. - Non-equilibrium steady states. - Non-locality and non-isotropy in diffusions and mean-field interactions.



A 5-day workshop at BIRS provides the perfect venue for an event like this. The group size of around 40 people is large enough for bringing together some of the main actors and several young researchers from the different involved communities, and compact enough for an optimal cross pollination of ideas.



The workshop will also provide a unique opportunity for young researchers to get a broad overview of the field and interact substantially with senior fellows. The number of talks will be limited to provide plenty of time for scientific discussion. We will prioritize younger researchers in our scientific program to give them the opportunity to explain their results to the present worldwide experts while we will schedule longer talks in the morning devoted to overviews on the new advances pointed above.



We are not aware of any other event focusing in this interdisciplinary aspect of entropies and nonlinear flows that is planned or being organized in the near future.