Scientific

Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.

We discuss various behaviours of continuous time simple random walks which are governed by the volume growth of the underlying weighted graph. In this setting the volume growth is computed with respect to a metric adapted to the random walk and not the graph metric. Use of these metrics allows us to establish results for graphs which are analogous to those for diffusions on a manifold or the Markov process associated with a strongly local Dirichlet form.

Consider a symmetric random walk on a group G. If the trace of the random walk generates G as a semigroup almost surely, then we say that G is algebraically recurrent. In this talk, we will present some initial steps towards understanding algebraic recurrence, including examples of algebraically recurrent and non-algebraically recurrent groups. We will conclude with some open questions. This is joint work with Itai Benjamini and Romain Tessera.

Here I survey a broad range of recent computational models for 2D and 3D cell motility. Some of these models depict chemical activation on the perimeter of a (static or deforming) domain. Others consider fluid and/or mechanical elements and/or biochemical signalling on the interior of a deforming 2D region
representing a cell. Examples of platforms include the immersed-boundary method and level set methods. I describe some of the computational challenges and how these have been addressed by various researchers.

Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.