Human behavioural patterns are fundamentally subject to inertia: the longer we have been participating in certain behaviours, lived in our current neighbourhoods, or even suffered under an addiction, the more difficult change becomes. This same concept has also been observed in the increasing political polarisation we have experienced in the past few years. We will discuss how renewal theory can be applied to these problems, with a focus on identifying strongly inertial states and the predictions associated with these. In particular, we will focus on empirically informed human migration or movement patterns, and recidivism models. That is, the likelihood of individuals to re-offend after a previous crime. A hallmark of strongly inertial systems is the lack of an equilibrium state, whereby their evolution is explicitly time dependent. These problems can be understood either through the lens of random walks or survival analysis.

Suppose you wish to use a blood test to screen a group of people for a rare disease. You could take a blood sample from each person and test the samples individually. However, it can be more efficient to mix a number of samples together and test that mixture: if the test comes back negative then none of those people have the disease, while if the test is positive then at least one of them has the disease and further investigation is needed. This problem is called group testing: given n people of whom k have the disease, how many of these mixed tests do we need to find out which people are infected? In this talk, we discuss recent progress on this question, concentrating on nonadaptive testing, where the tests are all designed in advance, so they can be conducted in parallel. We will look at practical algorithms and compare their performance to information theoretic limits.

Consider a system of $N$ particles moving according to Brownian motions and branching at rate one. Each time a particle branches, the particle in the system furthest from the origin is killed. It turns out that we can use results about a related free boundary problem to control the long term behaviour of this particle system for large $N$.

This is joint work with Julien Berestycki, Eric Brunet and James Nolen.

In the analysis of recursive algorithms and related random trees, fixed-point arguments involving the contraction method have proved fruitful over the last 30 years. I will speak about functional extensions of such results covering the analysis of classical tree-type data structures with nice geometric representations, extensions to random fields and connections to decompositions of random continuum real trees. The methods also allow to make statements on geometric properties of real trees including their fractal dimension and degrees. Examples include Aldous' CRT as well as dual trees of triangulations introduced by Curien and Le Gall.