The Patlak-Keller-Segel PDEs describe the chemotaxis of slime mould, and exhibit finite-time blow-up when the system mass is above a certain threshold--a mathematical manifestation of the biological observation that mould aggregates when its quantity is sufficiently high. In this talk, we associate a colliding stochastic particle system with these PDEs. By relating the squared Bessel process to the evolution of localized clusters of particles, we develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping; and apply this method to the numerical solution of the Patlak-Keller-Segel system and its variants. We show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. (Joint with Ibrahim Fatkullin.)

Gleb completed his PhD in math at the University of Arizona. He then spent two years as a postdoc in the School of Maths at the University of Edinburgh. He's interested in applications of probability and statistics in areas such as evolution, intracellular transport, and particle systems.