I will present a collection of moment computations over the unitary, symplectic and special orthogonal matrix ensembles that I've done throughout my thesis. I will focus on the methods used, the motivation from number theory, the relationship to Painlev\'e equations, and directions for future work.

In recent years, our understanding of the asymptotic behavior of characteristic polynomials of random matrices has seen much progression. A key paradigm in this area is that the asymptotic behavior is often captured by an appropriate family of Gaussian multiplicative chaos (GMC) measures (defined heuristically as the normalized exponential of log-correlated random fields). Indeed, such results have been shown for Harr distributed matrices for U(N), O(N), and Sp(2N), as well as for one-cut Hermitian invariant ensembles (and in particular, GUE(N)). In this talk we explain an extension of these results to GOE(2N) and GSE(N). The key tool is a new asymptotic relation between the moments of the characteristic polynomials of all three classical ensembles. 

I will present a study of integrable structures and quantum chaos in a class of infinite-dimensional though computationally tractable models, called quantum resonant systems. These models, together with their classical counterparts, emerge in various areas of physics, such as nonlinear dynamics in anti-de Sitter spacetime, but also in Bose-Einstein condensate physics. The class of classical models displays a wide range of integrable properties, such as the existence of Lax pairs, partial solvability or generic chaotic dynamics. This opens a window to investigate these properties from the perspective of the corresponding quantum theory by effectively diagonalising finite-sized matrices and exploring level spacing statistics. We will furthermore analyse the implications of the symmetries for the spectrum of resonant models with partial solvability and discuss how the rich integrable structures can be exploited to constructed novel quantum coherent states that effectively capture sophisticated nonlinear solutions in the classical theory.

Modern atmospheric and ocean science require sophisticated geophysical fluid dynamics models. Among them, stochastic partial

differential equations (SPDEs) have become increasingly relevant. The stochasticity in such models can account for the effect

of the unresolved scales (stochastic parametrizations), model uncertainty, unspecified boundary condition, etc. Whilst there is an

extensive SPDE literature, most of it covers models with unrealistic noise terms, making them un-applicable to

geophysical fluid dynamics modelling. There are nevertheless notable exceptions: a number of individual SPDEs with specific forms

and noise structure have been introduced and analysed, each of which with bespoke methodology and painstakingly hard arguments.

In this talk I will present a criterion for the existence of a unique maximal strong solution for nonlinear SPDEs. The work

is inspired by the abstract criterion of Kato and Lai [1984] valid for nonlinear PDEs. The criterion is designed to fit viscous fluid

dynamics models with Stochastic Advection by Lie Transport (SALT) as introduced in Holm [2015]. As an immediate application, I show that 

the incompressible SALT 3D Navier-Stokes equation on a bounded domain has a unique maximal solution.

This is joint work with Oana Lang, Daniel Goodair and Romeo Mensah and it is partially supported by European Research Council (ERC)

Synergy project Stochastic Transport in the Upper Ocean Dynamics (https://www.imperial.ac.uk/ocean-dynamics-synergy/

Nonuniform Ellipticity is a classical topic in PDE, and regularity of solutions to nonuniformly elliptic and parabolic equations has been studied at length. I will present some recent results in this direction, including the solution to the longstanding issue of the validity of Schauder estimates in the nonuniformly elliptic case obtained in collaboration with Cristiana De Filippis. 

It the talk, various conditions of local regularity of axisymmetric suitable weak solutions, including the so-called slightly supercritical ones, will be discussed.

The soap bubble theorem says that a closed, embedded surface of the Euclidean space with constant mean curvature must be a round sphere. Especially in real-life problems it is of importance whether and to what extent this phenomenon is stable, i.e. when a surface with almost constant mean curvature is close to a sphere. This problem has been receiving lots of attention until today, with satisfactory recent solutions due to Magnanini/Poggesi and Ciraolo/Vezzoni.
The purpose of this talk is to discuss further problems of this type and to provide two approaches to their solutions. The first one is a new general approach based on stability of the so-called "Nabelpunktsatz". The second one is of variational nature and employs the theory of curvature flows. 

Most of SDE models in epidemics, ecology, biology, finance etc. are highly nonlinear and do not have explicit solutions. Monte Carlo simulations have played a more and more important role. This talk will point out several well-known numerical schemes may fail to preserve the positivity or moment of the solutions to SDE models. Reliable numerical schemes are therefore required to be designed so that the corresponding Monte Carlo simulations can be trusted. The talk will then concentrate on new numerical schemes for the well-known stochastic Lotka--Volterra model for interacting multi-species. This model has some typical features: highly nonlinear, positive solution and multi-dimensional. The known numerical methods including the tamed/truncated Euler-Maruyama (EM) applied to it do not preserve its positivity. The aim of this talk is to modify the truncated EM to establish a new positive preserving truncated EM (PPTEM).