Mathematics

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.

The study of maps, that is of graphs embedded in surfaces, is a popular subject that has implications in many branches of mathematics, the most famous aspects being purely graph-theoretical, such as the four-color theorem. The study of random maps has met an increasing interest in the recent years. This is motivated in particular by problems in theoretical physics, in which random maps serve as discrete models of random continuum surfaces. The probabilistic interpretation of bijective counting methods for maps happen to be particularly fruitful, and relates random maps to other important combinatorial random structures like the continuum random tree and the Brownian snake. This course will survey these aspects and present recent developments in this area.

N.B. Due to microphone problems, the audio at the beginning of this recording is poor.

It is well known that a random walk in d>2 dimensions where the steps are i.i.d. mean zero and fully supported (not restricted to a hyperplane), is transient. Benjamini, Kozma and Schapira asked if we still must have transience when each step is chosen from either μ1 or μ2 based on the past, where μ1 and μ2 are fully supported mean zero distributions. (e.g. we could use μ1 if the current state has been visited before, and μ2 otherwise). We answer their question, and show the answer can change when we have three measures instead of two. To prove this, we will adapt the classical techniques of Lyapunov functions and excessive measures to this setting. No prior familiarity with these methods will be assumed, and they will be introduced in the talk. Many open problems remain in this area, even in two dimensions. Lecture based on joint work with Serguei Popov (Campinas) and Perla Sousi (Cambridge).

Mechanics of invadopods