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

[@ig2d]

Which is held in the highest esteem - the Abel prize or the fields medal?

[@akhiljalan11]

There is no universal answer since the prizes have different goals and criteria. The Fields Medal is restricted to those under 40 years and designed to "encourage" rising mathematical talents; of course, all of the winners are very talented and arguably have achieved a lifetime's worth of results by age 40 already. The Abel Prize is typically awarded to those much older and in recognition of an entire lifetime's work. For example, recent Abel laureates include: 2023: Luis Caffarelli (born 1948) 2022: Dennis Sullivan (born 1941) 2021: Avi Wigderson (born 1956) & Laszlo Lovasz (born 1948) Note that while some Fields medalists go on to win the Abel prize as well (Milnor, Deligne) this is not always so. In my opinion the Abel prize deserves higher regard, simply because it takes into account a mathematician's cumulative contributions over a decades-long career. Also, many contributions have deep impacts which are only realized decades after their publication. For example, Avi Wigderson is a computer scientist; while he has been working since the 1980s, the impact of computer science on pure mathematics is only beginning to be appreciated today, 4 decades later.