In this talk, we will show how sharp bounds on the moments of the solutions to some stochastic heat equations can lead to various qualitative properties of the solutions. A major part of the method consists of approximating the solution by “independent quantities”. These quantities together with the moments bounds give us sharp almost sure properties of the solution.

The study of the Geometry of random nodal domains has attracted a lot of attention in the recent past, in particular due to their connection with famous conjectures such as Yau's conjecture on the nodal volume of eigenfunctions of the Laplacian on compact manifolds, and Berry's conjecture on the relation between the geometry of the nodal sets associated to these eigenfunctions and the geometry of the nodal sets associated to toric random waves.

At first, the randomness involved in the definition of random nodal domains is often chosen of Gaussian nature. This allows in particular the use of explicit techniques, such as Kac--Rice formula, to derive the asymptotics of many observables of interest (nodal volume, number of connected components, Leray's measure etc.). In this talk, we will raise the question of the universality of these asymptotics, which consists in deciding if the asymptotic properties of random nodal domains do or do not depend on the particular nature of the randomness involved. Among other results, we will establish the local and global universality of the asymptotic volume associated to the set of real zeros of random trigonometric polynomials with high degree.

We shall explain, from variational point of view, why the  Laplaciian operator introduced by Ebin-Marsden using deformations is suitable to describe the fluid motion in a milieu with viscosity.

TBC

In this talk I will present three families of differential equations (SDEs, BSDEs and PDEs) and their links to each other. The novel fact is that some of the coefficients are generalised functions living in a fractional Sobolev space of negative order. I will discuss the appropriate notion of solution for each type of equation and show existence and uniqueness results. To do so, I will use tools from analysis like semigroup theory, pointwise products, theory of function spaces, as well as classical tools from probability and stochastic analysis. The link between these equations will play a fundamental role, in particular the results on the PDE are used to give a meaning and solve both the forward and the backward stochastic differential equations.  

Stochastic differential equations have Taylor expansions in terms of iterated Wiener integrals. The convergence of such expansion depends on the limiting behavior of the order-N iterated integrals as N tends to infinity. Recently, there has been increased interests in processes stopped at a random time. A breakthrough in the study of the iterated integrals of Brownian motion up to the exit time of a domain was included in the work of Lyons-Ni (2012). The paper leaves open an interesting question: what is the sharp rate of decay for the expected iterated integrals up to the exit time. We will review the state of the art in this problem and report some recent progress. Joint work with Ni Hao (UCL).

I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness Theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. The result will be presented in its most general form where actions are not necessarily free and hence I will also introduce the notion of stratified spaces.

The story of human progress is often described as a succession of ‘eras’ or ‘ages’ that are characterised by their most dominant technologies (e.g., the bronze age, the industrial revolution or the information age). In modern times, the fast pace of technological progress has accelerated the succession of eras. In addition, the increasing complexity of inventions has made the task of determining when eras begin and end more challenging, as eras are less about the dominance of a single technology and more about the way in which different technologies are combined. We present a data-driven method to determine and uncover technological eras based on networks and patent classification data. We construct temporal networks of technologies that co-appear in patents. By analyzing the evolution of the core-periphery structure and centrality time-series in these networks, we identify periods of time dominated by technological combinations which we identify as distinct ‘eras’. We test the performance of our method using a database of patents in Great Britain spanning a century, and identify five distinct eras.

Many studies of digital communication, in particular of Twitter, use natural language processing (NLP) to find topics, assess sentiment, and describe user behaviour.
In finding topics often the relationships between users who participate in the topic are neglected.
We propose a novel method of describing and classifying online conversations using only the structure of the underlying temporal network and not the content of individual messages.
This method utilises all available information in the temporal network (no aggregation), combining both topological and temporal structure using temporal motifs and inter-event times.
This allows us to describe the behaviour of individuals and collectives over time and examine the structure of conversation over multiple timescales.