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Frontiers in kinetic theory: connecting microscopic to macroscopic scales - KineCon 2022

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

Programme
4th January 2022 to 24th June 2022
Organisers: 
Jose A Carrillo University of Oxford
Jacob Bedrossian University of Maryland, College Park
Jingwei Hu Purdue University
Clément Mouhot University of Cambridge

Scientific Committee:

Russel E. Caflisch (Courant-USA), Mihalis Dafermos (Cambridge), Dalibard Anne-Laure (Sorbonne Université-Paris), Mario Pulvirenti (Rome I), Sylvia Serfaty (Courant-USA), Endre Suli (Oxford), Eitan Tadmor (Maryland)

Programme Theme:

Kinetic theory dates back to Maxwell and Boltzmann and uses a statistical viewpoint to describe the dynamics of many-particle systems. Over the past decades, it has become an important area of pure and applied analysis, in particular for bridging microscopic and macroscopic descriptions of complex systems. On one hand, it is at the centre of fascinating developments in theoretical analysis of nonlinear PDEs, optimal transport, and metric geometry. On the other hand, numerical simulations of kinetic equations and modeling using a kinetic approach have also become ubiquitous in many science and engineering disciplines, due to the indispensable role of kinetic theory in the multiscale modeling hierarchy. Furthermore, kinetic theory has also been a cross-road of interactions with several areas of mathematical physics such as general relativity, plasma physics, and the quantum many-body problem. This programme aims to expand the knowledge in analysis, modelling, and numerics of kinetic theory and to foster the interactions between different communities.

From the theoretical analysis perspectives, we hope to address the following key questions: 1) Advance the mathematical study of Landau damping (i.e. phase mixing) and related effects in plasmas and similar models. These effects are known to be crucial to understanding how information is transferred to small scales in phase space in the kinetic theory of plasmas but much remains to be understood mathematically, especially in physically relevant settings such as those involving magnetic fields or collisions; 2) Advance the progress in many-particle limits such as the Boltzmann-Grad limit and the mean-field limit for long range interactions; 3) Investigate the invariant measures and relative entropy principles for particle descriptions; 4) Use the gradient flow approach to study spatially homogeneous kinetic equations; 5) Further develop the theory of nonlocal equations in general, such as the Boltzmann and the Landau equation.

From the modeling and numerical analysis perspectives, the following aspects will be explored: 1) Phase transitions arise in many applications of social sciences and math biology. We will leverage bifurcation theory, entropy methods, gradient flows, and probabilistic tools to study these phenomena. 2) Many problems in analysis of big data like clustering in graphs, ensemble Kalman filtering, and computational neuroscience models share certain commonalities: they are based on large networks of complicated dynamical systems for which methods of kinetic theory are applicable. 3) Solutions to kinetic equations possess many important physical properties/structures, e.g., positivity, entropy, gradient flows or hypocoercivity. We will advance the progress in development of structure-preserving numerical methods for multiscale kinetic equations, as well as the associated stability and convergence analysis. 4) Kinetic equations are high-dimensional, nonlinear and nonlocal. Development of fast and accurate methods for the Boltzmann and Vlasov-Landau equations still presents one of the central problems in kinetic simulations. We will explore both Eulerian and Lagrangian based methods and also some low rank representations. 5) Kinetic theory connects microscopic particle descriptions and macroscopic fluid descriptions. Benefiting from recent advances in scientific machine learning, we seek new tools to conduct model and dimension reduction as well as moment closure.

Fostering new collaborations and finding solutions to some of the open problems, conjectures, and applications suggested above is the main objective of the programme. We will bring together some of the best researchers working in the new frontiers of kinetic theory as well as the neighboring communities such as fluid mechanics, probability, computational science and other applied fields, such as plasma physics and multi-agent collective behavior, with whom new bridges are being built.

 

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons