Entropy Methods, PDEs, Functional Inequalities, and Applications (14w5109)
The proposed workshop is in part intended as a follow-up to the workshops ``Nonlinear diffusions: entropies, asymptotic behavior and applications'' and ``Nonlinear Diffusions and Entropy Dissipation: From Geometry to Biology", held at BIRS in April 2006 and in May 2010, respectively. These two workshops have been an extraordinary boost for the field, with a tremendous impact on a large scientific community. They brought together PDE analysts, geometers and probabilists which have been developing collaborations afterwards. A startling example is how cutting-edge entropy-based techniques were applied to problems of mathematical biology following the stimulating exchanges at the workshops.
Our new proposal is motivated by the rapid progress in the field and in particular by the emergence of new applications and connections, some of which have been sketched above. The proposed workshop will focus on these recent developments and disseminate the manifold novel ideas in the ``core communities'' (PDE analysts, differential geometers and probabilists) that are involved in entropy methods and among researchers from applied mathematics that use entropies as a tool. The potential speakers have been selected accordingly: there are established representatives from each of the ``core communities'' as well as applied mathematicians (many from the steadily growing field of mathematical biology). Moreover, we have taken care to include a reasonable number of young researchers, who will benefit from this great opportunity to get introduced to the broad spectrum of applications, learn about up-to-date techniques and establish collaborations.
From the current state of the art, we expect that the following emerging topics will play an important role in the scientific program of the workshop:
- generalizations of the entropy method to systems of evolution equations, especially in connection with cross-diffusion models in biology;
- entropy-based techniques to capture blow-up phenomena in mean-field equations with non-local interactions;
- evolution equations whose qualitative behaviour is related to the interplay between two different Lyapunov functionals, like in the success story of the Keller-Segel model;
- unifying approaches that foster the links between e.g.\ gradient flow structures and the theory of large deviations or concentration of measure, which are expected to have a significant impact on the foundation of the entropy method;
- discrete entropy methods, which is a novel line of research with some striking first results and many open problems, and with possible applications in numerical analysis.
From previous experience, we know that a 5-day workshop at BIRS provides the perfect environment to hold an event of the proposed type. 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 communities working on entropy methods, and it is small enough to create the right stimulating atmosphere to foster collaborations and establish new interactions. No other event of a comparable scientific content --- discussing the entire spectrum of entropy methods --- has been held recently or is planned for the near future.