Nonlinear generalizations of the Schrödinger equation are of wide applicability to a range of areas including atomic and optical systems, 
plasma physics and water waves.  In this  talk we revisit some principal excitations in atomic and optical systems (such as Bose-Einstein condensates and photo-refractive crystals), namely dark solitonic fronts in single-component systems, and dark-bright waves in multi-component systems. Upon introducing them and explaining their existence and stability properties in one spatial dimension, we will extend them both in the form of stripes and in that rings in two-dimensions, presenting an alternative (adiabatic-invariant based) formulation of their stability and excitations. We will explore their filamentary dynamics, as well as the states that emerge from their transverse (snaking) instability. Then, we will consider these structures even in three dimensions, in the form of planar, as well as spherical shell wave patterns and generalize our adiabatic invariant formulation there. Finally, time permitting, we will give some glimpses of how some of these dynamical features in 1d and 2d generalize in a multi-orbital, time-dependent quantum setting.

The flow of a thin film down an inclined plane is an important physical phenomenon appearing in many industrial applications, such as coating (where it is desirable to maintain the fluid interface flat) or heat transfer (where a larger interfacial area is beneficial). These applications lead to the need of reliably manipulating the flow in order to obtain a desired interfacial shape. The interface of such thin films can be described by a number of models, each of them exhibiting instabilities for certain parameter regimes. In this talk, I will propose a feedback control methodology based on same-fluid blowing and suction. I use the Kuramoto–Sivashinsky (KS) equation to model interface perturbations and to derive the controls. I will show that one can use a finite number of point-actuated controls based on observations of the interface to stabilise both the flat solution and any chosen nontrivial solution of the KS equation. Furthermore, I will investigate the robustness of the designed controls to uncertain observations and parameter values, and study the effect of the controls across a hierarchy of models for the interface, which include the KS equation, (nonlinear) long-wave models and the full Navier–Stokes equations.

The Navier-Stokes paradigm does not capture thermal fluctuations that drive familiar effects such as Brownian motion and are seen to be key to understanding counter-intuitive phenomena in nanoscale interfacial flows.  On the other hand, molecular simulations naturally account for these fluctuations but are limited to exceptionally short time scales. A framework that incorporates thermal noise is provided by fluctuating hydrodynamics, based on the so-called Landau-Lifshitz-Navier-Stokes equations, and in this talk we shall exploit these equations to gain insight into nanoscale free surface flows.  Particular attention will be given to flows with topological changes, such as the coalescence of drops, breakup of jets and rupture of thin liquid films for which both analytic linear stability results and numerical simulations will be presented and compared to the results of molecular dynamics.

A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs.  Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,...,n}.  Not just any function can occur as the speed of hereditary graph property.  Specifically, there are discrete ``jumps" in the possible speeds.  Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollob\'{a}s, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized.  In contrast to this, many aspects of this problem in the hypergraph setting remained unknown.  In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds.  The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss.  This is joint work with Chris Laskowski.

We take a look at difference fields with several commuting automorphisms. The theory of difference fields with one distinguished automorphism has a model companion known as ACFA, which Zoe Chatzidakis and Ehud Hrushovski have studied in depth. However, Hrushovski has proved that if you look at fields with two or more commuting automorphisms, then the existentially closed models of the theory do not form a first order model class. We introduce a non-elementary framework for studying them. We then discuss how to generalise a result of Kowalski and Pillay that every definable group (in ACFA) virtually embeds into an algebraic group. This is joint work in progress with Zoe Chatzidakis and Nick Ramsey.

We will be discussing a Fourier-analytic approach
to optimal matching between independent samples, with
an elementary proof of the Ajtai-Komlos-Tusnady theorem.
The talk is based on a joint work with Michel Ledoux.

Consider a random N by N unitary matrix chosen according to Haar measure. A classical result of Diaconis and Shashahani shows that traces of low powers of this matrix tend in distribution to independent centered gaussians as N grows. A result of Johansson shows that this convergence is very fast -- superexponential in fact. Similar results hold for other classical compact groups. This talk will discuss analogues of these results for N by N matrices taken from a classical group over a finite field, showing that as N grows, traces of powers of these matrices equidistribute superexponentially. A little surprisingly, the proof is connected to the distribution in short intervals of certain arithmetic functions in F_q[T]. This is joint work with O. Gorodetsky.