# Scientific

## Volume growth and random walks on graphs

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.

## Cut points for simple random walks

We consider two random walks conditioned “never to intersect” in Z^2. We show that each of them has infinitely many `global' cut times with probability one. In fact, we prove that the number of global cut times up to n grows like n^{3/8}. Next we consider the union of their trajectories to be a random subgraph of Z^2 and show the subdiffusivity of the simple random walk on this graph.

## Algebraic recurrence of random walks on groups

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.

## Mathematical Cell Biology Summer Course Student Lecture 11

Cell-cell signaling between macrophages and mammary tumor cells

## Mechanical Simulations of Cell Motility

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.

## Polymer Size Distributions (continued)

We continue the discussion from last time, and solve the polymer size distribution equations, which are linear in the case of constant monomer level.

In a distinct case, when monomer is depleted, we show that the size distribution evolves in two phases, where in the first, the entire distribution appears to satisfy a transport equation, and then, later on, once monomer is at its critical level, the process of length adjustment appears to be governed by

an effective diffusion (in size-class). Next, I introduce the problem of determining features of polymer assembly from experimental

polymerization versus time data. (Based on work by Flybjerg et al, this leads to an extended homework exercise carried out by the students.) Finally, I revisit microtubule growth and shrinking by discussing the Dogterom-Leibler equations and their steady state exponential solutions. I illustrate the use of XPP software to solve several problems in this lecture.

## Interacting Particle Systems 2

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.

## Mathematical Cell Biology Summer Course Student Lecture 10

Diffusion of receptors on a cell membrane

## Random Maps 1

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.

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## Interacting Particle Systems 1

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.