If we fix a rectangle in the affine real space and if we choose at random a real polynomial with given degree d, the probability P(d) that a component of its vanishing locus crosses the rectangle in its length is clearly positive. But is P(d) uniformly bounded from below when d increases? I will explain a positive answer to a very close question involving real analytic functions. This is a joint work with Vincent Beffara.

 

The Ising model is one of the most classical statistical mechanics model, which has seen spectacular mathematical and physical developments for almost a century. The description of its scaling limit at the phase transition is at the center of a fascinating (conjectured) connection between statistical mechanics and field theories. I will discuss how recent mathematical progress allows one to make the connection between the two-dimensional Ising model and Conformal Field Theory rigorous. If time allows, I will discuss the insight this gives one into related models and field theories.

Based off joint works with S. Benoist, D. Chelkak, H. Duminil-Copin, R. Gheissari, K. Izyurov, F. Johansson-Viklund, K. Kytölä, S. Park and S. Smirnov

If a dynamical system has a conservation law, i.e. a constant along the trajectory of the motion, the study of its evolution along the trajectories of a perturbed system becomes interesting. Conservation laws can be seen everywhere, especially at the level of probability distributions of a reduced dynamic.  We explain this with a number of models, in which we see a singular perturbation problem and identify a conservation law, the latter is used to seek out the correct scale to work with and to reduce the complexity of the system. The reduced dynamic consists of a family of  ODEs with rapidly oscillating right hands side from which in the limit we obtain a Markov process. For stochastic completely integrable system, the limit describes the evolution of the level sets of the family of Hamiltonian functions over a very large time scale.

The Yang-Mills heat equation is the gradient flow corresponding to the Yang-Mills functional. It was initially introduced by S. K. Donaldson to study the existence of irreducible Yang-Mills connections on the projective plane. In this talk, we will consider this equation over compact three-manifolds with boundary. It is a nonlinear weakly parabolic equation, but we will see how one can prove long-time existence and uniqueness of solutions by gauge symmetry breaking. We will also demonstrate some strong regularization results for the solution and see how they lead to detailed short-time asymptotic estimates, as well as the long-time convergence of the Wilson loop functions. 

One of the challenges of 21st-century science is to model the evolution of complex systems.  One example of practical importance is urban structure, for which the dynamics may be described by a series of non-linear first-order ordinary differential equations.  Whilst this approach provides a reasonable model of urban retail structure, it is somewhat restrictive owing to uncertainties arising in the modelling process.

We address these shortcomings by developing a statistical model of urban retail structure, based on a system of stochastic differential equations.   Our model is ergodic and the invariant distribution encodes our prior knowledge of spatio-temporal interactions.  We proceed by performing inference and prediction in a Bayesian setting, and explore the resulting probability distributions with a position-specific metrolpolis-adjusted Langevin algorithm.

Ambitious mathematical models of highly complex natural phenomena are challenging to analyse, and more and more computationally expensive to evaluate. This is a particularly acute problem for many tasks of interest and numerical methods will tend to be slow, due to the complexity of the models, and potentially lead to sub-optimal solutions with high levels of uncertainty which needs to be accounted for and subsequently propagated in the statistical reasoning process. This talk will introduce our contributions to an emerging area of research defining a nexus of applied mathematics, statistical science and computer science, called "probabilistic numerics". The aim is to consider numerical problems from a statistical viewpoint, and as such provide numerical methods for which numerical error can be quantified and controlled in a probabilistic manner. This philosophy will be illustrated on problems ranging from predictive policing via crime modelling to computer vision, where probabilistic numerical methods provide a rich and essential quantification of the uncertainty associated with such models and their computation. 

Identifying correlations within multiple streams of high-volume time series is a general but challenging problem.  A simple exact solution has cost that is linear in the dimensionality of the data, and quadratic in the number of streams.  In this work, we use dimensionality reduction techniques (sketches), along with ideas derived from coding theory and fast matrix multiplication to allow fast (subquadratic) recovery of those pairs that display high correlation.

Joint work with Jacques Dark

 I will give a light introduction to the theory of regularity structures and then discuss recent developments with regards to renormalization within the theory - in particular I will describe joint work with Martin Hairer where multiscale techniques from constructive field theory are adapted to provide a systematic method of obtaining needed stochastic estimates for the theory. 

Abstract: Equations with small scales abound in physics and applied science. When the coefficients vary on microscopic scales, the local fluctuations average out under certain assumptions and we have the so-called homogenization phenomenon. In this talk, I will try to explain some probabilistic approaches we use to obtain the first order random fluctuations in stochastic homogenization. If homogenization is to be viewed as a law of large number type result, here we are looking for a central limit theorem. The tools we use include the Kipnis-Varadhan's method, a quantitative martingale central limit theorem and the Stein's method. Based on joint work with Jean-Christophe Mourrat. 

Wave propagation in random media can be studied by multi-scale and stochastic analysis. We first consider the direct problem and show that, in a physically relevant regime of separation of scales, wave propagation is governed by a Schrodinger-type equation driven by a Brownian field. We study the associated moment equations and clarify the propagation of coherent and incoherent waves. Second, using these new results we design original methods for sensor array imaging when the medium is randomly scattering and apply them to seismic imaging and ultrasonic testing of concrete.