The Junior Applied Mathematics Seminar (JAMS) is a weekly seminar with talks given by junior researchers on a topic in Applied or Industrial Mathematics. It is a great opportunity to share research, get feedback from other students and to become more confident in presenting work. We also provide coffee and cake!

This year, there are two different types of talks:

*Classic JAMS:* Short talks, somewhat specialised and more focused on new results. A great opportunity to get know what fellow PhD students’ research is about.

*Introductory JAMS:* Longer talks, but more accessible and educational. These are meant to introduce you to the basic ideas of active research fields, and give you a broader picture of modern Applied Mathematics.

If you are interested in giving a talk (of either type) or have further questions, please email one of the committee members.

You can also sign up to our __Mailing List__ to receive regular updates on these events!

*Venue (for October):* For the first month of the term we will be meeting in Huxley 340, every Wednesday at 4 pm.

**Upcoming Seminars**

**(22/03/17) Arshad Kamal (Imperial College London): “Translational and rotational correlations for swimming through heterogeneous media”
4.00 pm in LT 340, Huxley**

Many existing studies for swimming through complex fluids assume a microstructure with a length scale much smaller than that of the swimmer. Our study explores the case where they are comparable. The swimmer would thus see the surrounding medium as a set of obstacles in a viscous fluid, rather than a continuum.

We present our model of a simple undulatory swimmer moving through an environment of obstacles that are tethered to random points in space via linear springs. By varying the obstacle density and tether stiffness, we investigate how the mechanical interactions with the obstacles alter the swimming behavior.

We find that the swimming velocity, and its fluctuations, can vary non-monotonically with obstacle density, for a fixed stiffness. This is observed to be also true for the fluctuations in the angular velocity.

Interestingly, we find that the velocity and the angular velocity of the swimmer are correlated. There is an obstacle density for which this correlation can be maximized, for a fixed tether stiffness. We explain why this correlation is to be expected in the context of our swimmer and obstacle configuration.

**Past Seminars**

**(15/03/17) Matthew Garrod (Imperial College London): “Spatial Network Models and Their Applications”
4.00 pm in LT 340, Huxley**

Many complex systems can be represented as networks or graphs. These include social systems, transport systems and even networks of neurons in the brain. A large number of these systems are embedded in space or have some metric structure where physically closer nodes are more likely to share a connection. I will discuss some of the primary models of spatial networks and some of their applications.

I will also discuss some recent work on quantifying the variability of the dynamical properties of these networks and the factors which influence our ability to predict these properties.

**(01/03/17) Benjamin Walter (Imperial College London): “On the first-passage-time problem of the Ornstein Uhlenbeck process”
4.00 pm in LT 340, Huxley**

**Abstract:** The Ornstein-Uhlenbeck-process, a stochastic process describing a particle in a harmonic potential subject to thermal noise, finds application in many different areas of research. We use simple and coupled Ornstein-Uhlenbeck processes to model lifetimes of cellular junctions in biological tissue to understand how mechanical properties such as elasticity of a tissue depend on external parameters. Therefore, we studied the first-passage-time-problem of the Ornstein Uhlenbeck process, i.e. the probability distribution of the time it takes the particle to travel from A to B, in the limit of small distances. We apply different analytical methods to show that the probability distribution is governed by one dimensionless parameter that interpolates between two different regimes (spring-driven, noise-driven) and validate this finding with numerical simulations. We then discuss exact and effective models for non-harmonic and coupled systems and difficulties that arise in each case.

**(22/02/17) Adam Gosztolai (Imperial College London): “Optimal adaptation in bacterial chemotaxis”
4.00 pm in LT 340, Huxley
**

Many microorganisms are capable of navigating chemical attractant landscapes, a behaviour known as chemotaxis, with the objective to actively move towards higher concentrations. I will begin my talk explaining the classical approach to modelling the chemotaxis of E. coli, one of the most well studied systems in biophysics. I will compare agent-level and coarse-grained approaches pointing out some of the discrepancies, particularly when cells navigate fluctuating gradients. This will motivate the introduction of a novel modelling approach inspired by optimal control theory, that can account for some of the previously unexplained behaviour. Equipped with the new model, I will explore some of the predictions that can shed light on how bacterial cells ‘learn’ and exploit their environments.

**(1/02/17) Hemant Khatri (Imperial College London): “Why is weather prediction so difficult?”
**

**4.00 pm in LT 340, Huxley**

**(1/02/17) Nikolas Nusken (Imperial College London): “Accelerating convergence to equilibrium for stochastic dynamics”**

**4.00 pm in LT 340, Huxley**

**(25/01/17) Maximilian Engel (Imperial College London): “Killed stochastic processes, quasi-stationary distributions and dynamical systems”**

4.00 pm in LT 340 (40 min), Huxley

The topic goes back to the pioneering work by Yaglom in 1947 but in recent years new ideas have been developed. We discuss concepts like quasi-stationary distributions and quasi-ergodicity, referring to an example from population dynamics. If time, we will discuss our work relating these concepts to dynamical systems with holes and local random dynamical systems.

**(18/01/17) Cezary Olszowiec (Imperial College London): “Topological methods for differential equations”**

4.00 pm in LT 340 (40 min), Huxley

I will briefly recall some of the most commonly used tools from the qualitative theory of differential equations and present examples of their applications, e.g. to the models from celestial mechanics and neuroscience.

**(7/12/16) Jure Vogrinc (Imperial College London): “Introduction to MCMC”**

4.00 pm in LT 340, Huxley

**(30/11/16) Marina Ferreira (Imperial College London): “A framework for modelling packed cell tissues”**

4.00 pm in LT 340, Huxley

In some cell tissues the cells are tightly packed with almost no intercellular space. The force each cell exerts on its neighbors can not be neglected, as it plays an important role in the mechanics of the tissue. We propose a mathematical framework consisting of a geometric representation of individual cells which interact with other cells aiming at minimizing a local potential under non-overlapping constraints. This gives rise to a non-linear non-convex minimization problem for which there are not many tools available. In this talk, I will show an algorithm we have developed for this type of problems: the damped Arrow-Hurwicz algorithm, as well as some examples and numerical results. This framework can be very useful in the study of packed tissues, as it helps to predict the impact of intercellular forces on the dynamics of the whole tissue, which can not be easily addressed through laboratory experiments.

**(30/11/16) Jordan Noble (Imperial College London): “Correlation-based Streaming Anomaly Detection in Cybersecurity”**

4.00 pm in LT 340, Huxley

Methodology for the statistical analysis of enterprise network data is becoming more important in cybersecurity. The volume and velocity of enterprise network data sources puts a premium on streaming analytics – procedures that pass over the data once, while handling temporal variation in the process. This talk will introduce a procedure for streaming anomaly detection in the correlation between a pair of variables. This procedure is intended to detect anomalous behaviour in the traffic between a pair of devices, represented by an edge in the network graph. The approach is illustrated on real Netflow data, where novel ideas are introduced to assess performance on a single edge. The procedure is then successfully extended to combine and score anomalies across multiple edges, localised in time and network space.

**(23/11/16) Michael Hartl (Imperial College London): “The ODE method for stochastic approximation”**

**4.00 pm in LT 340, Huxley
**We will introduce stochastic approximation processes in the rather general form of the Robbins-Monro algorithm and then explore the so called ODE approach, invented by Ljung et al., as a method of describing the asymptotic behavior of such a process. This will include a generous review of several concepts from ODE theory, like chain recurrence and pseudotrajectories, as they were introduced by Conley in 1978. Finally we will apply our newfound insights to an example with strong links to a recent historical event.

**(23/11/16) Johannes Pausch (Imperial College London): “Tears of wine and other phase transitions ”**

**4.00 pm in LT 340, Huxley**

When you drink a lot of wine, you might have noticed that, sometimes, wine can crawl up on the inner side of the glass until larger drops are formed and flow back as tears into the the sea of wine at the bottom the glass. Interestingly, you don’t have to be drunk to see this effect. This, and many other effects are caused by the interplay of three things: gravity, solid-liquid interaction, surface tension. Surprisingly, we are still at the start of understanding and predicting these phenomena. In my talk, I will explain the basics of classical liquid layers in equilibrium and give a few examples of interesting phenomena.

**(16/11/16) Josh Plasse (Imperial College London): “Data Streams and Adaptive Estimation”**

**4.00 pm in LT 340, Huxley
**Data streams are potentially unending sequences of ordered observations that are subject to unknown temporal variation. The mining of these streams poses several challenging problems to the statistics and machine learning communities. This talk will introduce data streams and discuss why performing inference on them is a non-trivial task. We then discuss a framework which allows for temporally adaptive parameter estimation for well-known distributions.

**(16/11/16) Calum Skene (Imperial College London): “The effect of mean flow swirl on the transfer function of an M-flame ”**

**4.00 pm in LT 340, Huxley**

Combustion instabilities are a major problem in industry today, affecting all sorts of machinery from gas turbines to propulsion systems. The use of swirl injectors is becoming increasingly common, therefore it is important to fully understand the mechanisms leading to instability and the role increased swirl plays in stabilization. Presented here is a linear numerical study of the optimal forcing of a steady M-flame. The optimal long-time response to harmonic forcing will be computed for different azimuthal wave-numbers and temporal frequencies, with an emphasis put on how increases in swirl change the dynamics.

The optimal forcing will be found using a singular value decomposition (SVD) technique. It will be shown how to efficiently and accurately obtain the linear and adjoint operators from a non-linear code. As each run of a SVD calculation is computationally expensive, it is important to squeeze out all the possible information from a single run. To this end, we will derive gradient information from the singular triplets, enabling us to better grasp the dynamics and carry out parametric sensitivity for a small overhead.

**(09/11/16) Sam Brzezicki (Imperial College London): “Solving 2D Stokes Flow problems using Complex Variables: An Introduction”**

**4.00 pm in LT 340, Huxley
**Nearly all fluid dynamicists will know about the complex potential, w(z), used to study ideal flow and in particular how powerful it can be in providing solutions to fluids problems. So, for those studying low-Reynolds number flow, why do we not learn about something analogous?

In this talk I’ll outline how we can introduce complex variables to study low-Reynolds number flows and I’ll show how we can use the beautiful theory of complex analysis to solve physical problems.

This talk should not only be of interest to those studying fluids or those that want to learn about low-Reynolds number flow, but also to any mathematician that loves the complex plane!

**(09/11/16) Arshad Kamal (Imperial College London): “Micro-scale Undulatory Locomotion in Heterogeneous Media”**

**4.00 pm in LT 340, Huxley
**Many microorganisms swim in non-Newtonian, complex fluids. A lot of the models for swimming through such fluids assume that the environment microstructure has a much smaller length scale than that of the swimmer.

There are, however, notable examples where the microstructure is at the same length scale as the swimmer. In this case, the swimmers experience the surrounding medium as a set of obstacles suspended in a viscous fluid, rather than a continuum.

We study such a situation for a simple undulatory swimmer as it moves through an environment of obstacles that are tethered to random points in space via linear springs. Our simulations are based on the force-coupling method, a technique used to study suspensions of particles in Stokes flow.

We examine how swimming behavior is altered by mechanical interactions with the obstacles by varying obstacle density and spring stiffness. We find that the mechanical interactions can either enhance or hinder locomotion, and often for fixed stiffness, there is an obstacle density for which the average speed is maximized.

In addition, we find that the velocity fluctuations are also highly dependent on environment composition and a non-monotonic dependence can be found here as well.

**(02/11/16) Matt Price-Williams (Imperial College London): “Detecting cyber attacks by identifying statistical anomalies”**

**4.00 pm in LT 340, Huxley**

In recent years the number of cyber attacks on computer networks has increased where intruders may be looking to exfiltrate valuable data. This talk aims to introduce statistical approaches to cyber-security, in particular identifying anomalous behaviour in computer networks that could represent an intruder traversing the network.

**(02/11/16) Thomas Helfer (King’s College London): “Black hole formation from axion stars”**

**4.00 pm in LT 340, Huxley**

The classical equations of motion for an axion possesses quasi-stable, localized, oscillating solutions, which we refer to as “axion stars”. We study, for the first time, collapse of axion stars numerically using the full non-linear Einstein equations of general relativity and the full non-perturbative cosine potential. We map regions on an “axion star stability diagram”. We identify three regions of the parameter space: i) long-lived oscillating axion star solutions, modulated by self-interactions, ii) collapse to a BH and iii) complete dispersal due to gravitational cooling and interactions. We discuss the observational relevance of our findings for axion stars as BH seeds, which are supermassive in the case of ultralight axions.

**CANCELLED (26/10/16) Jure Vogrinc (Imperial College London): “Introduction to MCMC”**

**4.00 pm in LT 340, Huxley**

Markov chain Monte Carlo (MCMC) algorithms, such as Metropolis-Hastings algorithm or the Gibbs sampler, have become indispensable tools for numerical computation of high dimensional integrals as well as for sampling from complicated probability distributions. I this introductory talk I would like to explain what MCMC algorithms are, why they are important, talk briefly about issues the theory is facing and show some distinct examples of their usage.

**(19/10/16) Gerard McCaul (King’s College London): “How to win friends and influence functionals”**

**4.00 pm in LT 340, Huxley
**Path integrals are interesting. Occasionally they are even useful. One example of the latter is in the context of open quantum systems, where the system’s interaction with a heat bath is explicitly modelled. Many useful results can be obtained only through the use of influence functionals—a clever trick used to re-express a very general interacting quantum system in terms of a single-particle stochastic differential equation. Here we present a simple derivation of the Feynman-Vernon influence functional and its various uses. This leads to an exact (!) quantum Langevin equation following results by Kleinert, Shubanov and Stockburger. Finally we will explore the consequences of generalising the initial conditions by expressing them as a path integral, and the effect this has on our equations of motion. This promises to be no fun for either the presenter or the audience.

**(19/10/16) Alessandro Milazzo (Imperial College London): “The Italian Pension Problem: a Stochastic Optimal Control Approach”**

**4.00 pm in LT 340, Huxley**

Stochastic optimal control is a mathematical procedure used to find optimal solutions when dealing with a stochastic decision problem. As stated by the name, the decision problem is governed by a control (usually a function of time) that can be set in order to optimize the objective function. I will show the mathematical framework standing behind the formulation of the problem, and the theorems and steps that lead to the optimal solution.

Many of these optimization problems belong to the financial and the actuarial fields. Interested by these techniques, I applied this approach to a problem concerning the Italian pension system.

I built a model that mirrors the aforementioned pension problem and I used stochastic optimal control techniques to find the optimal investment strategy. The closed-form solutions are presented and some Monte Carlo simulations are carried out to show how the optimal investment strategy varies with respect to parameters variation.

**(12/10/16) Fabrizio Bianchi (Imperial College London): “What is the Mandelbrot set?”**

**4.00 pm in LT 340, Huxley
**In this introductory talk I would like to give an idea of what is holomorphic dynamics, i.e., the study of dynamical systems arising from the iteration of holomorphic maps. Starting from the very basics and providing a fair amount of examples, I will introduce Fatou and Julia sets, their main properties, and then explain the relation between the Mandelbrot set and the stability and bifurcations of such dynamical systems.

**(12/10/16) Alexis Arnaudon (Imperial College London): “On a new integrable PDE for strongly correlated Bose-Einstein condensates”
4.00 pm in LT 340, Huxley
**This talk will be a quick journey through a few topics in mathematics and physics. I will start in the physical world of Bose-Einstein condensates (BECs) that can be mathematically understood with PDEs such as the nonlinear Schrödinger equation (NLS).

In this well developed topic I will take a sharp turn to a more recent phenomenon that arises if the BECs are strongly correlated. In this case a large number of atoms evaporates and one needs a new equation to describe these atoms. This equation was recently derived by M. Kira and is called the hyperbolic Bloch equation (HBE). After coupling this HBE with the NLS in order to describe the dynamics of the entire system, I will apply some approximations that will produce the so-called hyperbolic reduced Maxwell-Bloch equation (hRMB). From here, I will leave the physics and study the mathematics of this new equation which has the remarkable property to be completely integrable. I will discuss this property and show some explicit solutions that may be observable in strongly correlated BECs. This is a joint work with John D. Gibbon, and is available on arXiv with number 1609.00690.

**(04/10/2016) Maxime Morariu-Patrichi (Imperial College London): “Limit Order Book Modelling With Interacting Point Processes” **

**2.30 pm in LT 340, Huxley
**Nowadays, on more than half of financial markets, trading happens through limit order books. In the first part of my talk, I introduce the limit order book mechanism along with typical quantities such as the bid and ask prices. Using real data, I give some summary statistics, enabling the audience to gain a better intuition about the market dynamics. Moreover, I show an animation of a limit order book, both in real time and slow motion. In the second part, I discuss the limit order book modelling problem and present two main modelling trends: Markov queues and Hawkes processes. I focus on the core ideas behind each model type as well as on their conceptual limitations. I end the talk by briefly explaining the specific problem that I am working on.

**(04/10/2016) Jake Dunn (Imperial College London): “Bootstrapping and Differential Equations ”
2.30 pm in LT 340, Huxley
**The purpose of this talk to discuss some methods that are used in non-linear differential equations.

I will work through the framework for dealing with some Non-linear ODEs, discussing: local existence, uniqueness and global existence. The techniques used will be that of: contraction maps, Gronwall’s lemma, energy methods and bootstrapping. This framework serves as a basis for the analysis PDEs (far more complicated equations). The talk aims to highlight in detail the concept of bootstrapping. Bootstrapping is a novel way of proving boundedness out of seemingly nothing and can be thought of as a continuous extension of proof by induction. It is an invaluable tool used heavily in the analysis of dispersive PDEs.

**(22/04/2016) Sergio Perez: “Collective Behaviour Models: How do fish in a school align and stay together?” Huxley 747, 12:30 h**

**(Imperial College London / Technical University of Madrid)**

Abstract: Emergent phenomena of self-organized particles such as flocking, crowd, and swarming behaviours have recently received considerable attention due to its engineering, physical, and biological applications. Our project is devoted to study some of these collective behaviour models mainly from a hydrodynamic point of view, although microscopic and mesoscopic models will also be treated.

In this talk we will focus our attention on the Cucker-Smale model, which basically describes alignment behaviour. It was introduced in 2007 and since then a lot of research has been carried out. Some interesting properties of these systems will be discussed, and a special consideration will be given to the numerical methods needed to simulate them. In addition, different potentials will be added to the system in order to evaluate their effects.

**(28/04/2016) Nils Caillerie: “When the cane toads attack : a non-Darwinian evolution process for aggressive invasion” Huxley 711c, 16:00 h**

**(Ecole Normale Superieure de Lyon)**

Abstract: Australia’s Eastern coast is threatened by the propagation of an invasive species of toads. This invasion has drawn the attention of the naturalists sine cane toads at the edge of the front are more dispersive and they share this behaviour with their offspring. This means that this behaviour is a phenotypical manifestation of a genetic trait, and that we are in presence of a non-Darwinian evolution process, led by a spatial sorting rather than adaptive considerations.

Thanks to data on the trajectories of toads collected by a team of Australian naturalists, we will try to understand the motion dynamics of those toads and challenge the biologists hypothesis on what this genetical trait might be. We will test two mathematical models (reaction-diffusion equations and kinetic equations) and see why we should follow the second lead to match the data.

**Past Seminars**

Spring 2016: [Schedule] [Archive]

Autumn 2015: [Schedule] [Archive]

Spring 2015: [Schedule] [Archive]

Autumn 2014: [Schedule] [Archive]