How does the number of friend groups change as more people join a social network? Are individuals with a mutual popular friend more likely to be friends with each other? How can we apply this to non-social settings?

During this talk we will investigate results for the asymptotic expected number of cliques in the Chung-Lu Inhomogeneous Random Graph Model (in which nodes are independently assigned "popularity" weights with tail probabilities $h^{1−a}l(h)$, where $a>2$ and $l$ is a slowly varying function) as the graph grows, the probability of triangles given one node's popularity, and how to interpret those results in a social context.

Lymphocyte populations, stimulated in vitro or in vivo, grow as cells divide. Stochastic models are appropriate because some cells undergo multiple rounds of division, some die, and others of the same type in the same conditions do not divide at all. If individual cells behave independently, each can be imagined as sampling from a probability density of times to division. The most convenient choice of density in mathematical and computational work, the exponential density, overestimates the probability of short division times. We consider a multi-stage model that produces an Erlang distribution of times to division, and an exponential distribution of times to death. The underlying idea is to divide the cell cycle into a given number of stages, and the cell is required to sequentially visit each stage in order to divide. At each stage, each cell may either proceed to the next one or die. Cells can be classified across generations depending on the number of times that they have undergone cell division, and the interest is in estimating the number of cells in each generation over time, which can be then compared to appropriate experimental data. Using Approximate Bayesian Computation based on Sequential Monte Carlo (ABC-SMC) methods, we compare our model to published cell counts, obtained after CFSE-labelled OT-I and F5 T cells were transferred to lymphopenic mice. The death rate is assumed to scale linearly with the generation and the number of stages of undivided cells (generation 0) is allowed to differ from that of cells that have divided at least once (generation greater than zero). Multiple stages are preferred in posterior distributions, and the mean time to first division is longer than the mean time to subsequent divisions. Our multi-stage model is able to account for competition between cellular fates (cell death vs division) while incorporating a non-exponential division time probability distribution, and allows us to find closed expressions for the mean number of cells in each generation.

This is joint work with Martín López-García, Grant Lythe, and Carmen Molina-París.

In this talk I shall discuss the tail behaviour of Gaussian multiplicative chaos and explain precise asymptotics of the leading order under mild assumptions. At criticality, the leading order coefficient is fully explicit in all dimensions and does not depend on any local variation of the underlying field, demonstrating an interesting universality phenomenon.

Stochastic quantisation is a method introduced in the physics literature by Nelson and Parisi-Wu in order to quantise Euclidean field theories. Its basic principle is to view a quantum field as the invariant measure of a Langevin dynamic. In this talk, I will review some recent progress of this method for gauge theories from the side of mathematics. I will in particular describe a state space for the 3-dimensional quantum Yang-Mills theory and an associated Markov process for which the Yang-Mills measure is conjecturally invariant.

Based on arXiv:2201.03487, which is joint work with Ajay Chandra, Martin Hairer, and Hao Shen.