Abstract:
Emergence is a phenomenon of formation of collective outcomes in systems where communication between agents has local range. For a wide range of applications, such as swarming behavior of animals or exchange of opinions between individuals, such outcomes result in a globally aligned state or congregation of aligned clusters. The classical result of Cucker and Smale states that alignment is unconditional in flocks that have global communication with nonintegrable radial tails. Proving a similar statement for purely local interactions is a challenging mathematical problem. In this talk we will overview three programs of research directed on understanding the emergent phenomena: statistical approach to generic alignment for agentbased systems, kinetic approach based on relaxation and hypocoercivity, and hydrodynamic models incorporating a novel way of interaction based on topological communication.

Roman Shvydkoy is a professor of mathematics at University of Illinois at Chicago. His research interests include fluid dynamics, Euler, NavierStokes systems; Nonlocal parabolic equations; Collective behavior, emergent dynamics, flocking; Turbulence, Onsager conjecture, convex integration, vanishing viscosity limit; Instability in ideal fluids, linearization method, WKB analysis, geometric optics; Spectral theory, pseudodifferential, harmonic and functional analysis.

This lecture was part of the biannual Abel Symposium.

This year the title of the symposium was Partial Differential Equations waves, Nonlinearities and Nonlocalities.

The symposium was funded by
The Norwegian Academy of Sciences and Letters via the Abel board and The Norwegian Mathematical Society
NTNU Norwegian University of Science and Technology
Research Council of Norway via the grant Waves and Nonlinear Phenomena
Trond Mohn Foundation