The talk will contain a review of several recent results on the analytic continuation of the random analytic functions. We will start from the classical theorem on the random Taylor series (going to Borel’ s school), but the main subject will be the random zeta – function (which was introduced implicitly by Cramer) and its generalizations. We will show that “true primes are not truly random “, since zeta – functions for the random “pseudo-primes” (in the spirit of Cramer) have no analytic continuation through the critical line Re (z) = 1/2.

Viewed forwards in time, a population reproducing according to some random mechanism can be thought of as a branching process. What if it is viewed backwards? We can take a sample of individuals from the current generation and trace their genealogy backwards, and for instance find their most recent common ancestor; this is known as a coalescent process. If we know a population's random mating process, but have no actual data as to what the phylogenetic tree looks like, how do we derive the distribution of the time until its most recent common ancestor? I will discuss a variant on the classical Wright-Fisher reproductive model and deduce some parameter thresholds for emergence of different qualitative features of the tree. An isomorphic problem may also be useful in computer science for bounding the running time of certain random sampling algorithms.

In this talk I will review the basic ideas of the regularity structure theory developed by Martin Hairer, as well as its applications to stochastic PDE problems. I will then discuss my joint work with Hairer on the sine-Gordon equation and central limit theorems for stochastic PDEs.

Markov chain methods for Monte Carlo simulation of complex physical or statistical models often require significant tuning. Recent theoretical progress has renewed interest in "adaptive" Markov chain algorithms which learn from their sample history. However, these algorithms produce non-Markovian, time-inhomogeneous, irreversible stochastic processes, making rigorous analysis challenging. We show that lower bounds on the mixing times of these processes can be obtained using familiar ideas of hitting times and conductance from the theory of reversible Markov chains. The bounds obtained are sufficient to demonstrate slow mixing of several recently proposed algorithms including adaptive Metropolis kernels and the equi-energy sampler on some multimodal target distributions. These results provide the first non-trivial bounds on the mixing times of adaptive MCMC samplers, and suggest a way of classifying adaptive schemes that leads to new hybrid algorithms. Many open problems remain.

Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

Quantum field theory is the theoretical framework for studying fundamental interactions. "Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. The recent years witnessed interesting progress in understanding solutions of these stochastic PDEs, one of the remarkable examples being Hairer and Mourrat-Weber's results on the Phi^4_3 equation.

In this talk we will discuss stochastic quantization of quantum field theory with gauge symmetries, with focus on an Abelian example but also provide prospects of non-Abelian Yang-Mills theories. We address issues regarding Wilson’s lattice regularization, dynamical gauge fixing, renormalization, Ward identities, and construction of dynamical loop and string observables.

I'll discuss work on shift invariance in a half space setting. These are non-trivial symmetries allowing certain observables of integrable models with a boundary to be shifted while preserving their joint distribution. The starting point is the colored stochastic six vertex model in a half space, from which we obtain results on the asymmetric simple exclusion process, as well as for the beta polymer through a fusion procedure, both in a half space setting. An application to the asymptotics of a half space analogue of the oriented swap process is also given.

The injective norm is a natural generalization to tensors of the operator norm of a matrix. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, where it is known as the geometric entanglement. We give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, corresponding to a lower bound on the geometric entanglement of random quantum states. The proof is based on spin-glass methods, the Kac—Rice formula, and recent progress coming from random matrices. Joint work with Stéphane Dartois.