The 1^{st} Early Career Applied Mathematics Meeting will be held online, via zoom, on Thursday, March 18^{th} 2021, between 12:45 and 17:30 GMT. I am organising this meeting jointly with Elena Luca (UCL). Please email us if you would like to attend this meeting but have not received a zoom link.

The meeting is rather loosely organised around the theme of "Complex and numerical analysis". We have six invited talks, half-hour each. There is also a plenary talk at the end of the meeting. Professor Demetrios Papageorgiou (Imperial College London) will be delivering the plenary talk.

You can find the flyer and program for the meeting below or you can download it here. Scroll down this page for the abstracts of the talks.

"Waves on the microscale: Order, chaos and its control"

I will give an overview of the type of mathematics that needs to be invoked to study nonlinear waves in small scale geometries. Applied mathematicians are very familiar with large scale waves, such as water waves, and the activities that emanate from familiar models like the celebrated Kortweg de-Vries equation (a google search of "Kortweg de-Vries" comes up with 650,000 hits!). At large scales things are gravity driven and viscosity plays a secondary role and can be ignored. On the microscale, however, gravity is typically diminished and viscosity rules. Interfaces between immiscible fluids (i.e. waves) are quite happy to stay uniform and trundle along in their viscous morass. To do engineering on the microscope we need to drive them out of their equilibrium. One way to do this is by using external electric or magnetic fields, and I will begin with an overview of the mathematical models that emerge from such interventions - they involve a crucial coupling between the Navier-Stokes equations and the Maxwell equations in the right limit. The result is a host of PDEs that are derived asymptotically. Interestingly, these PDEs can produce chaotic solutions (we have rigorous proofs of this) even at zero Reynolds numbers. After deriving some of the models I will present computations of their solutions (mostly in the form of movies) and also address theoretically the problem of control and optimal control of such systems showing that this is possible and opens a gateway to useful physical exploitations.

"A current-valued solution of the Euler equation and its application"

Besides the Euler flow, there is another mathematical model that describes 2D incompressible and inviscid fluid flows, called the point vortex dynamics, which is derived formally from the Euler flow. Since the point vortex dynamics is a finite-dimensional Hamiltonian system, it is mathematically easier to handle this than the Euler flow. For this reason, the point vortex dynamics is of importance in applications, since it is utilized as simple models for fluid phenomena with localized vortex structures. In the meantime, since the point vortex dynamics is derived formally from the Euler flow, it still remains open whether the insight gained by using the point vortex dynamics is also applicable to the Euler flow as well. Therefore, in order to solve this problem, we need to establish that the point vortex dynamics is an Euler flow in a mathematically appropriate sense. In other words, the problem is to set up a space of solutions to the Euler equation that contains the pair of the fluid velocity and the pressure for the point vortex dynamics. However, since the vorticity of the point vortex dynamics is given by a linear combination of delta functions, due to the singularity of the vorticity and the nonlinearity of the Euler equation, the problem of mathematical justification is considered to be a difficult one in the analysis of the 2D Euler equation. Although there have been many studies to date on the indirect characterization of the point vortex dynamics by approximate sequences of weak solutions to the Euler equation, the problem remains open in terms of constructing a space of solutions that directly includes the point vortex dynamics. In this talk, we will discuss the problem is solved by a geometric method, rather than an analytic one.

"Charged elastic materials: a variational view point"

We prove the existence of minimizers for a model describing a charged nonlinearly elastic material, characterized by a coupling between nonlinear elastic energy and capacitary term defined on the deformed configuration (Eulerian). Under additional geometric assumptions on limiting deformed sets, we show continuity of the capacity under uniform convergence of deformations.

"Diagonalising the infinite: How to compute spectra with error control"

Spectral theory is ubiquitously used throughout the sciences to solve complex problems. This is done by studying "linear operators", a type of mapping that pervades mathematical analysis and models/captures many physical processes. Just as a sound signal can be broken down into a set of simple frequencies, an infinite-dimensional operator can be decomposed (or "diagonalised") into simple constituent parts via its spectrum (the generalisation of eigenvalues). Often spectra can only be analysed computationally, and computing spectra is one of the most investigated areas of applied mathematics over the last half-century. Wide-ranging applications include condensed-matter physics, quantum mechanics and chemistry, fluid stability, optics, statistical mechanics, etc. However, the problem is notoriously difficult. Difficulties include spectral pollution (false eigenvalues of finite-dimensional approximations/truncations that "pollute" the true spectrum) and spectral inexactness (parts of the spectrum may fail to be approximated). While there are algorithms that in certain exceptional cases converge to the spectrum, no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and (c) provides approximate eigenvectors. This may lead to incorrect simulations in applications. It has been an open problem since the 1950s to decide whether such reliable methods exist at all. We affirmatively resolve this question, and we prove that the algorithms provided are optimal, realising the boundary of what computers can achieve. Moreover, the algorithms are easy to implement and parallelise, offer fundamental speed-ups, and allow problems to be tackled that were previously out of reach, regardless of computing power. The method is applied to difficult physical problems such as the spectra of quasicrystals (aperiodic crystals with exotic physical properties).

"Dirichlet-to-Neumann map for evolution PDEs on the half-line with time-periodic boundary conditions"

Download this abstract here.

"The theory of interacting aerofoils"

When two or more aerofoils move together their interactions can significantly affect the characteristics of the surrounding flow. Indeed, it is well-known that many natural fliers and swimmers exploit these effects to enhance their propulsive efficiency. This raises the question of when interacting aerofoils operate in co-operation or competition i.e. when do the interaction effects help or hinder the aerofoils' motion. We develop a rigorous mathematical theory for these interactions using conformal maps, multiply connected function theory and modified Schwarz problems. Via use of the transcendental Schottky--Klein prime function, our theory is valid for any connectivity (any number of aerofoils). Accordingly, our approach is very general and permits consideration of a range of wing motions (pitching, heaving, undulatory) and configurations (tandem, in-line, periodic, ground effect). We also develop a fast numerical implementation of our theory and thus explore the physical implications of interaction effects. We focus on the (doubly connected) case where there are two interacting swimmers and find that our theory yields excellent agreement with experimental data. Specifically, we recover the equilibrium configurations observed in recent experiments and postulate the existence of new stable configurations. In summary, our theory is the natural generalisation of the classical thin aerofoil theory that dominated early aerodynamics research.

"Hollow vortex in a corner"

A point vortex near a wall will propagate parallel to the wall. In 1895, Pocklington generalized this solution by finding the shape of a propagating hollow vortex moving parallel to a wall. In 2013, Crowdy, Llewellyn Smith, and Freilich re-expressed Pocklington's solution using the Schottky--Klein prime function. Here we use the prime function to find the boundary of a hollow vortex at rest in straining flow in a corner of arbitrary angle. We construct the solution using conformal maps from a canonical doubly connected annular domain to the physical plane. The result is a two-parameter family of solutions depending on the corner angle and on the non-dimensional ratio of strain to circulation. Possible generalizations to domains of higher connectivity are discussed.