Random growth occurs in many real world settings, for example, we see it exhibited in the growth of tumours, bacterial growth and lightning patterns. As such we would like to be able to model such processes to determine their behaviour in their scaling limits. Well studied models to describe these different processes include the Eden model and DLA. In a 1998 paper, Hastings and Levitov introduced a one parameter family of conformal maps HL$(\alpha)$ which can be used to model Laplacian growth processes and allows us to vary between the previous models by varying the parameter $\alpha$. We will consider a regularised version of this model and analyse its scaling limits and fluctuations for different values of alpha under capacity rescaling.

In this talk, I will discuss some recent developments on the study of singular stochastic wave equations. I also describe some similarities and differences between stochastic wave and heat equations, indicating particular difficulty of the dispersive/hyperbolic problem.

Small connected subgraphs (such as edges and triangles) are important network summary statistics and play a role in many parts of network science, including analysis of biological networks and network comparison methodologies. Finding the distribution of the number of copies of such subgraphs in certain random graph models is therefore of interest. In this talk, we review classical results that concern the normal and Poisson approximation (depending on the parameter regime) of the number of copies of subgraphs in the Erdos–Renyi $G(n,p)$ model. We then consider a generalisation from the $G(n,p)$ model to the stochastic block model (with possibly multiple edges), and obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in these random graph models.

We demonstrate applications of discte-time Markov decision processes (MDPs) with two examples. In the first one, in absence of impulses, the system dynamics evolve deterministically according to a flow. The decision maker can apply impulses at any time moment, which act on the trajectories. The running cost is accummulated over time, in addition to the cost incurred each time an impulse is applied. The problem is to minimize the total (expected) cost, subject to some functional constraints of the same type. We obtain its solvability result by applying relevant results for discrete-time MDPs. The second one is similar to the first one, but the underlying system is a Markov pure jump process, and apart from impulse controls, the decision maker can also apply the gradual control on the local characteristics. We may reduce it to one with only gradual control, whose solvability result can be in turn obtained from those for discrete-time MDPs. For this reason, relevant facts about discrete-time MDPs will be reviewed.

Most methods to solve ill-conditioned or ill-posed inverse problems resort to regularisation techniques to make the problem well-posed. This often requires setting the value of the so-called regularisation parameters that control the amount of regularisation enforced. This talk presents a new empirical Bayesian method for setting regularisation parameters in high-dimensional inverse problems that are convex. A main novelty is that the parameters are calibrated by maximum marginal likelihood estimation, using a stochastic gradient algorithm that is driven by two proximal Markov chain Monte Carlo samplers, tightly combining modern optimisation and sampling techniques. The proposed methodology is demonstrated on several challenging inverse problems and compared to other techniques for setting regularisation parameters.

I will present two results related to random bases of a matroid: an asymptotically optimal mixing time bound for the bases-exchange random walk to sample bases of a matroid, and a concentration bound for Lipschitz functions over random bases of a matroid. Both results are consequences of a modified log-Sobolev inequality for r-homogeneous strongly log-concave distributions.
The proof is simple and elementary. No functional analysis is involved. I will present a key lemma which bounds the decay rate of relative entropy, implying all results above.

Joint work with Mary Cryan and Giorgos Mousa.