Past JAMS Seminars

When liquid-liquid interfaces are dealt with in mathematical modelling -and this occurs very often- a no-slip condition is imposed at their interface. This means that both liquids are expected to nicely flow “hand-in-hand” moving with the same interfacial velocity. The stability of such “well-behaved” liquids has been extensively studied to reveal an instability caused by their different viscosities (see for example Yih 1967, Hooper & Boyd 1983, Renardy 1985). But what happens when the liquids are rather naughty and prefer to have more fun, when they stop “holding hands” and start sliding on the interface? What happens to the stability of the system when they overdo it and this slippage goes to infinity? In this talk we will address the following questions:

 In this talk we will address the following questions:

🖊 Why is interfacial slip of interest?

🖊 How is this slip incorporated in the models?

🖊 What happens to the stability of the system when slip is present?

🖊 What happens when the liquids go crazy and the slip tends to infinity?

 Asymptotics, numerics and colourful graphs unite their powers in an attempt to provide some answers. See you there!

The van der Pauw method is commonly used in the applied sciences to find the resistivity of a simply connected, two-dimensional conducting laminate. Given the usefulness of this ‘4-point probe’ method there has been much recent interest in trying to extend it to holey, that is, multiply connected, samples. In this talk I will introduce two new mathematical tools to this area of investigation—the prime function on the Schottky double of a planar domain and the Fay trisecant identity—and uses them to show how the van der Pauw method can be extended to find the resistivity of a sample with a hole. I show that an integrated form of the Fay trisecant identity provides valuable information concerning the appearance of ‘envelopes’ observed in the case of holey samples by previous authors. I find explicit formulae for these envelopes, as well as an approximate formula relating two pairs of resistance measurements to the sample resistivity that is expected to be valid when the hole is sufficiently small and not too close to the outer boundary. I describe how these new mathematical tools have enabled us to prove certain conjectures recently made in the engineering literature.

In this talk I consider the interactions of electromagnetic waves under oblique incidence with arrays of metamaterial cylinders, or metacylinders. Metacylinders composed of a closely-spaced array of thin plates have previously been shown to exhibit anisotropic scattering, internal trapping and the tunable focusing of incident radiation. Herein I recast this problem in electromagnetism and consider the interaction of light under TE polarization with metacylinders of a metallic plate array microstructure, demonstrating the same aforementioned wave phenomena. I develop upon this scattering model and consider infinite and truncated one-dimensional gratings of metacylinders, or metagratings, of tunable microstructure angle. I retrieve the reflection and transmission spectra of plane-wave incidence upon the metagrating, before demonstrating the existence of guided Rayleigh-Bloch modes. In both cases I show how the tunability of the microstructure angle modifies the wave interaction, and in doing so note that the metagrating acts as a tunable surface that is invisble at certain angles, with a high degree of control over resonance and multiple scattering.

Heterogeneity at the single-cell level dramatically affects the behaviour of the entire cell population. The sources of this heterogeneity can be related to the stochasticity in the cell cycle duration or to different sizes or ages exhibited by the cells. Since the description of these scenarios is usually only amenable to simulations, in this talk I will develop a novel stochastic modelling approach in order to portray the role played by cell-to-cell variability in cell population dynamics. I show that the stochastic behaviour of a heterogeneous cell population can always be encoded in a nonlinear Volterra integral equation; this integral formulation allows us to obtain several insights about the cell communities such as extinction probabilities, lineages distribution, critical phases and stochastic moments of the abundances. My research proposes a novel methodology to describe the stochastic behaviour of cell-structured populations using numerical, computational and analytical methods. The results open a new theoretical path to understanding stochastic mechanisms underlying fluctuations in various biological and medical applications such as extinction of cancer cell populations under treatment, cell population growth in adverse environments and dormancy-awakening transition in breast cancer.

Accurately simulating the dynamics of the ocean comes with many challenges, one of which is the exceptional computational cost of running a global ocean model at high resolution. However, if we reduce the resolution then we lose small scale features, many of which play important roles in the dynamics. It is therefore necessary to parameterise these small scale effects. I shall explain how a method for doing this may be derived from a variational principle endowed with a stochastic part, which can be tuned to capture the effects of the missing scales.

Introduction to Cyber Security and Machine Learning
While standard malware analysis and defense lags behind continuous adversarial advancements, faster and smarter proactive measures are necessary. The latest buzz is to turn towards machine learning, but how can it help exactly and be guaranteed to work? I will highlight the potential advancement for cybersecurity at every step using machine learning and data science techniques. Multiple regressions can be harnessed for threat hunting, identifying major threats and assisting security specialists in prioritizing their analysis. Sifting through large quantities of network traffic logs can be simplified through deep learning, enriching the information with major threat characteristics. Models provide generic insights into our data, for current and future threats, so standard techniques can maximize their effectiveness.

Instead of talking about my work directly, I plan to give a brief tour of some of the methods that myself and many other people use to extract low-dimensional models from high-dimensional data. Arguably the most well-known of these are POD and DMD, along with their numerous variants. I will also introduce SINDy, a method which is able to infer the underlying equation from a dataset using sparse regression methods. While many of these methods were initially inspired by fluid dynamics, they have already been adopted in other fields such as financial mathematics and the modelling of infectious diseases.

The concept of a tipping point (or critical transition) describes a phenomenon where the behaviour of a physical system changes drastically, and often irreversibly, compared to a relatively small change in its external environment. A number of generic mechanisms have been identified which can cause a system to tip. One such mechanism is rate-induced tipping, where the transition is caused by a parameter changing too quickly – rather than moving past some critical value. In this talk I will explain the mathematics behind rate-induced tipping and illustrate the concept with an ecological example.

The term ‘undulatory movement’ refers to wave-like movement patterns that an organism (such as snakes, worms, cells etc.) may adopt in order to propel itself through its environment. This type of movement inherently lends itself to mathematical analysis. Developing models to establish the optimal undulatory movement pattern, and using these models as a phenotyping lens in C. elegans locomotion, the aim of my project is to establish the genetic underpinnings of gait efficiency, and why organisms move the way they do.

The study of dynamical systems have been established since the last century and they are many theoretical insight and theorems that are built for solving real-world problems. In particular, there are many studies conducted on the discrete dynamical system and the birth of chaos from seemingly simple systems by changing the parameters. However, most studies done are on the underlying deterministic system, but realistically there exists some errors and uncertainties which should be taken into account. Hence, we aim to investigate the minimal invariant set of the system with bounded noise and its bifurcation. A numerical approach will be taken to give insight into some critical problems faced when approximating these minimal invariant sets. We also investigate the simple case of a linear map and the dynamics involved on the boundary of the minimal invariant set.

The Boltzmann Equation describes the evolution of the probability density governing the position and velocity of a cloud of particles. If we discount interaction between particles, we get a linear equation which can be viewed abstractly as a (hyperbolic) transport operator, plus a non-local diffusive operator, with parabolic flavour. Which behaviour dominates depends on the cross-section, the (spatially-varying) coefficient of the parabolic operator. Very recently, this was quantified: to be precise, under what conditions on the cross-section are we guaranteed exponential speed of convergence to the equilibrium distribution? The condition derived is interesting, and the proof is elegant, relying on simple semigroup arguments.

In this talk, after a whistle-stop tour of the linear Boltzmann equation, we will construct a simple example with sub-polynomial convergence time. We’ll then give an overview of the proof, focussing on the condition derived, showing it is both necessary and sufficient for exponential convergence. Anyone with an interest in PDEs or stochastic particle models will hopefully find interest in this talk.

Acoustic metamaterials are artificial materials that possess remarkable capabilities to enhance, focus and guide the propagation of sound waves. These unique materials are built from periodic arrays of resonant building blocks, wherein the array spacing is much smaller than the incident wavelength so that the metamaterial can be described as an effective homogeneous medium. In practice, this means that the resonators must be subwavelength, exhibiting a resonance whose wavelength is considerably larger than its characteristic size. For acoustic metamaterials, the most commonly employed resonators are air bubbles and Helmholtz resonators, which are hollow cavities having small apertures. In this talk, I will show how one can develop asymptotic models for certain acoustic metamaterials by exploiting the several distinguished length scales of the system.

In this presentation, we discuss the effective diffusion coefficient associated with the generalized Langevin equation (GLE) in a periodic potential. We present formal asymptotic results in the small correlation time regime, as well as in the overdamped and underdamped limits. We also apply a recently developed numerical method in order to calculate the effective diffusion coefficient for a wide range of friction coefficients, confirming our asymptotic results and corroborating the findings of earlier studies on the subject. Finally, we study the convergence of the solution of the GLE to equilibrium using techniques from the theory of hypocoercivity.

In this joint work with Darryl Holm, we investigate the role of timescales in determining wave current interaction in the ocean. In particular, we will employ the variational framework to derive simplified models via asymptotic expansions and compare with the direct approach of asymptotic expansions in the equations of motion.

The Hamiltonian Monte Carlo (HMC) method for sampling measures is a bit like skateboarding: you push off with some initial momentum, slide down a ramp (or, the potential energy surface), stop, and repeat with a different initial momentum until you have explored the entire ramp (I will not demonstrate this). This algorithm and its generalisation, where the algorithm “remembers” the previous value of the momentum at each iteration, has become standard in the data science community due to its efficiency in sampling from very high dimensional space. In this talk, I will show how one can obtain the generalised HMC algorithm as some discretisation of a stochastic Hamiltonian system, and show how this idea can be used to construct an efficient and practical algorithm that can sample from Lie groups and not just vector spaces. Joint work with Alexis Arnaudon and Alessandro Barp.

Nonparametric extensions of topic models such as Latent Dirichlet Allocation, including Hierarchical Dirichlet Process (HDP), are often studied in natural language processing. Training these models generally requires use of serial algorithms, which limits scalability to large data sets and complicates acceleration via use of parallel and distributed systems. Most current approaches to scalable training of such models either don’t converge to the correct target, or are not data-parallel. Moreover, these approaches generally do not utilize all available sources of sparsity found in natural language – an important way to make computation efficient. Based upon a representation of certain conditional distributions within an HDP, we propose a doubly sparse data-parallel sampler for the HDP topic model that addresses these issues. We benchmark our method on a well-known corpora (PubMed) with 8m documents and 768m tokens, using a single multi-core machine in under three days.

In this talk I will investigate how the laminar-turbulent transition of a flow falling down a plane can be influenced by introducing flexibility. Starting from the Naiver-Stokes equations, we will derive a linear system to solve numerically. The techniques to get there will be briefly covered, and the results presented, ie. lots of nice, pretty pictures. Using the generality of our solution we can study both high and low viscosity fluids, comparing to analytical results.

We prove a version of the mountain pass theorem for lower semicontinuous and lambda-geodesically convex functionals on the space of probability measures P(M) equipped with the W_2 Wasserstein metric, where M is a (compact, complete, connected) Riemannian manifold or R^d. As an application of this result, we show that the empirical process associated to a system of weakly interacting diffusion processes exhibits a form of noise-induced metastability. The result is based on an analysis of the associated McKean–Vlasov free energy, which for suitable attractive interaction potentials has at least two distinct global minima at the critical parameter value b = b_c. Joint work with Andre Schlichting.

An estimated 250 million people live in areas less than 5m above average sea level. As sea levels rise and storms increase in strength and frequency due to climate change, the coastal zone is becoming an ever more critical location for the application of advanced mathematical techniques. There is also a high degree of uncertainty associated with these models.  By considering the case study of beach erosion, I will show how statistical methods can be used in conjunction with numerical models to quantify uncertainty. In particular, I will focus on the relatively new method of Multilevel Monte Carlo.

In this talk I will present an introduction to mean field games (MFGs) as a limit of stochastic differential games as the number of players goes to infinity. I will explain the notion of an equilibrium and discuss the PDE description of such an equilibrium.

In the second half of this talk I will develop an MFG model of innovation that we can use to evaluate the size of knowledge spillovers.

As a result of rainfall during the ascent or descent stages of the flight or de-icing operation, usually a thin layer of viscous liquid is formed or rain droplets are left above the aircraft wings. The presence of the droplet leads to significant changes in boundary-layer behaviour.

In this talk we are going to demonstrate a modified tripe-deck model which accounts for the presence of a droplet and its effects on the air flow. Analytical solutions are obtained for the linearised flow’s velocity field, pressure and skin friction.

There exists a two-way coupling between the troposphere and the stratosphere; atmospheric waves are generated through a variety of mechanisms in the troposphere and they propagate upwards in to the otherwise stable stratosphere. Here, they may break down or they may be reflected downwards back in to the troposphere, thereby influencing tropospheric dynamics and forming a closed loop two- way coupling. Placing this coupling in an appropriate mathematical framework is crucial to improving climate and weather predictions, and this is investigated by first understanding the reflection of Rossby waves.
The mechanism by which Rossby waves are reflected is studied in terms of a nonlinear critical layer. Critical layers exist where the base flow velocity equals the phase speed, and it is here that nonlinear effects may become significant, and Rossby waves reflected. This work takes in to account the presence of all harmonics within the critical layer, and attempts to identify the reflected and transmitted waves as well as the consequential disturbances within the critical layer using multiple-scale techniques.

In this talk, I will introduce basic Natural Language Processing ideas and the machine learning approaches that are generally used for dealing with the problem of text classification. After this I will highlight the particular challenges posed by this particular project.

“When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first.”- Werner Heisenberg.

It is thought that one transatlantic flight can add as much to your carbon footprint as a typical year’s worth of driving. Why are planes so inefficient? Well, part of the reason is turbulence, which causes drag on the aircraft. Therefore one of the most important questions to ask is how can we control the flow so turbulence is less prevalent? Not only could this save the aerospace industry millions of pounds in fuel, but it would also have a huge impact on the sustainability of air travel, and help reduce the dependence on oil which is expected to run out within the next century.

This talk will cover a brief introduction to the equations governing fluid flows, and the emergence and structure of turbulence in basic boundary-layer flows. Specifically, the talk will cover what structures emerge in the transition from laminar to turbulent flow and how these can be described mathematically. These structures can help to provide an understanding of the underlying ‘scaffold’ of turbulence. This knowledge could be implemented to reduce their prevalence and therefore reduce turbulence/drag on an aircraft.

Keywords: fluid dynamics, turbulence, boundary layer, laminar flow control

A mesh adaptive shallow water solver is created for solving tsunami modelling problems. This builds upon previous work at Imperial College in the Firedrake and Thetis projects. The former project provides efficient, finite element solving software and the latter specialises this for coastal, estuarine and ocean modelling on unstructured meshes. An anisotropic mesh adaptivity library is created which is capable of adaption both to fields related to the flow (such as free surface displacement) and as guided by adjoint solution data. Applying mesh adaptivity to shallow water problems, enables the efficient generation of numerical solutions to tsunami wave propagation problems. The case study of the tsunami which struck Fukushima, Japan, in 2011 is considered, wherein leading tsunami waves reached the coast in just ten minutes. Numerical results indicate computational cost can be reduced, whilst retaining a sufficiently high accuracy, under mesh adaptivity. Through establishing a highly efficient approach to numerical tsunami simulation, sufficient warning could be provided in future scenarios, allowing for evacuation and damage mitigation in those coastal areas determined to be most at risk.

The critical Rayleigh number $Ra_c$ for thermally convective instability depends on the wave-length of the disturbance.

In an annular spherical domain with separation d, there are degenerate points $(Ra_c , d _c )$ at which instability to two different sets of thermal-rolls occurs simultaneously.
This study provides a weakly non-linear analysis of the multiple-bifurcation problem, demonstrating that four distinct coupled amplitude equations govern the non-linear evolution of these interactions. The choice of which can be predicted from the inherent symmetry of the interacting modes. Considering a variety of $\ell:m$ mode interactions at different values of the Prandtl number $\sigma$, it is found that mixed mode solutions can exist only within certain regions of the parameter space. While for special resonant mode interactions a stable-period solution is found at low Prandtl number $\sigma$. In each case the weakly non-linear prediction is verified using direct numerical simulation.

We study the McKean-Vlasov equation on the flat torus which is obtained as the mean field limit of a system of interacting diffusion processes enclosed in a periodic box. The system acts as a model for several real world phenomena from statistical mechanics, opinion dynamics, collective behaviour, and stellar dynamics.

After commenting on the well-posedness of the equation, we study its long time behaviour and convergence to equilibrium. We then focus our attention on the stationary problem – under certain assumptions on the interaction potential, we show that the system exhibits multiple equilibria which arise from the uniform state through continuous bifurcations. This relates closely with previous work on phase transitions for the Mckean-Vlasov equation (cf. Chayes and Panferov, J. Stat. Phys., 2010). Finally, we attempt to classify continuous and discontinuous transitions for this system and show how this work, in conjunction with previous studies of the system, can be used to recover classical results on phase transitions for the noisy Kuramoto model. This is joint work with José Carrillo, Greg Pavliotis, and André Schlichting.

For long time accurate numerical integration, it is of paramount importance to preserve certain geometric structures
associated to the underlying system. This has been a key area of research in numerical analysis, known as geometric numerical integration, over the past few decades. In this talk we investigate the continuous Galerkin method as a geometric numerical integrator. We will also introduce a new discontinuous finite element method and discuss what geometric properties this method possesses.

We study a nonlinear reaction-diffusion system undergoing a logistic type competition. The diffusivity of species depends on the ratio of their population density and the resource distribution (nourishments). Such a feature provides us with not only a more realistic model of population dispersals but also interesting mathematical challenges. We will present some existence theorems together with a mollification strategy.

The countable random graph, also referred to as the Rado graph, is a very interesting mathematical object as it appears in many different areas of Maths. It can be constructed using ideas from probability and number theory (more precisely the Dirichlet’s Theorem). It possesses nice properties in logic, group theory, the Ramsey theory and topological dynamics. We will touch upon all these areas to see why the Rado graph makes a fascinating structure.

Abstract:
Many cells and microorganisms are motile: they are capable of navigating fluid environments to accomplish tasks required for survival or specialized functions. I will describe the physical principles obeyed by swimming at the micro scale and the mathematical models used to describe them. I will present numerical results for a sperm cell-like swimmer in an unbounded domain and compare several models of different fidelity based on the Stokes flow approximation. The models include a Regularised Stokeslet Method, a 3D Finite Element Method, the Resistive Force Theory versions of Lighthill and Gray and Hancock, as well as a simplified approximation based on computing the hydrodynamic forces exerted on the head and the flagellum separately. I will discuss applications that may arise from our ability of modeling and manipulating microswimmers and the challenges that lie ahead.

Biography:
Dr Cecilia Rorai earned her Ph.D. from the Doctoral School in Environmental and Industrial Fluid Mechanics, University of Trieste, Italy. She conducted her doctoral research at University of Maryland in collaboration with Professors D. P. Lathrop, R. M. Kerr and under the supervision of Professors K. R. Sreenivasan and Michael E. Fisher. She defended her thesis entitled “vortex reconnection in superfluid helium” in April 2012. Later, she moved as a visiting scientist at the National Center for Atmospheric Research (NCAR), Boulder, Colorado, where she worked on turbulence in stratified flows in collaboration with Professor Annick Pouquet, Dr D. Rosenberg and Professor P. D. Mininni. In 2013 she joined the Royal Institute of Technology (KTH) and the Nordic Institute of Theoretical Physics (NORDITA) Stockholm, Sweden, as the recipient of the postdoctoral fellowship from the Göran Gustafsson foundation. Her project dealt with the motion of elastic capsules in Stokes flows and involved Professor Luca Brandt, Dr Dhrubaditya Mitra and Dr Lailai Zhu. In 2016 she was awarded a Marie Skłodowska-Curie postdoctoral fellowship and she joined Queen Mary University where she works in collaboration with Dr Sergey Karabasov. Her current research focus is on biological fluid dynamics, more specifically, swimming at the micrometer scale. Her approach is theoretical and computational.

Abstract:
A triple-scale simulation of a molecular liquid is presented where atomistic, coarse-grained, and hydrodynamic descriptions of a liquid matter are consistently coupled. In the previous dual-scale model in the literature, which we will briefly overview, the continuum and discrete representations of the same substance are coupled together in the framework of conservation laws for mass and momentum that are treated as effective phases of a nominally two-phase flow. The effective phase distribution in space defines the computational domain decomposition for the low-resolution and the high-resolution zones of the model as defined by the user. The continuum representation is based on the Navier-Stokes Fluctuating Hydrodynamics (NS-FH) equations, which use stochastic fluxes to model the effect of the Brownian motion at the small scale. Building on the dual-scale model, which used the classical molecular dynamic (MD) dynamics as a model for the discrete state of the fluid, the current triple-scale multiscale model replaces the pure MD model with an adaptive resolution scheme (AdResS). AdResS is a state-of-the art discrete particle method developed in the literature, which smoothly connects the atomistic and the mesoscopic scales. The new combined AdResS-NS-FH model is shown to perform more accurately and be less sensitive to the calibration parameters compared to the previous dual-scale model when the high-resolution zone is made to move arbitrarily for a stationary uniform medium test. We also present test results to show how the combined triple-scale model can correctly preserve the amplitude and the phase of a high-frequency acoustic wave that propagates across the continuum hydrodynamic, the mesoscopic particle, and the microscopic particle layers of the hybrid computational domain.

Biography:
Jingyi Hu currently is a third year PhD student of Dr Sergey Karabasov in the School of Engineering and Materials Science, Queen Mary University of London. Prior to becoming a PhD student, Jingyi got the bachelor degree in applied mathematics from Central South University, China in 2014. Focusing on the multiscale modelling in hydrodynamics, her PhD project aims at developing a new triple-level system that smoothly connects atomistic, mesoscale and continuum representation of liquids in application to shear effects in complex fluids.