Roman Shvydkoy: Generic alignment conjecture and the problem of emergence of collective behavior

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 non-integrable 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 agent-based 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, Navier-Stokes systems; Non-local 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, pseudo-differential, harmonic and functional analysis.

This lecture was part of the bi-annual 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

Видео Roman Shvydkoy: Generic alignment conjecture and the problem of emergence of collective behavior канала The Abel Prize