Past Industrial and Applied Mathematics Seminar

2 February 2017
Neave O'Clery, Asbjorn Nilsen Riseth

What makes cities successful? A complex systems approach to modelling urban economies

Urban centres draw a diverse range of people, attracted by opportunity, amenities, and the energy of crowds. Yet, while benefiting from density and proximity of people, cities also suffer from issues surrounding crime, congestion and density. Seeking to uncover the mechanisms behind the success of cities using novel tools from the mathematical and data sciences, this work uses network techniques to model the opportunity landscape of cities. Under the theory that cities move into new economic activities that share inputs with existing capabilities, path dependent industrial diversification can be described using a network of industries. Edges represent shared necessary capabilities, and are empirically estimated via flows of workers moving between industries. The position of a city in this network (i.e., the subnetwork of its current industries) will determine its future diversification potential. A city located in a central well-connected region has many options, but one with only few peripheral industries has limited opportunities.

We develop this framework to explain the large variation in labour formality rates across cities in the developing world, using data from Colombia. We show that, as cities become larger, they move into increasingly complex industries as firms combine complementary capabilities derived from a more diverse pool of workers. We further show that a level of agglomeration equivalent to between 45 and 75 minutes of commuting time maximizes the ability of cities to generate formal employment using the variety of skills available. Our results suggest that rather than discouraging the expansion of metropolitan areas, cities should invest in transportation to enable firms to take advantage of urban diversity.

This talk will be based on joint work with Eduardo Lora and Andres Gomez at Harvard University.


Hamilton-Jacobi-Bellman equations for dynamic pricing

I will discuss the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear, second-order, terminal value PDE problem. The equation arises in optimal control theory as an optimality condition.

Consider a dynamic pricing problem: over a given period, what is the best strategy to maximise revenues and minimise the cost of unsold items?

This is formulated as a stochastic control problem in continuous time, where we try to find a function that controls a stochastic differential equation based on the current state of the system.

The optimal control function can be found by solving the corresponding HJB equation.

I will present the solution of the HJB equation using a toy problem, for a risk-neutral and a risk-averse decision maker.

  • Industrial and Applied Mathematics Seminar
26 January 2017
Mariano Beguerisse Díaz

Cells adapt their metabolic state in response to changes in the environment.  I will present a systematic framework for the construction of flux graphs to represent organism-wide metabolic networks.  These graphs encode the directionality of metabolic fluxes via links that represent the flow of metabolites from source to target reactions.  The weights of the links have a precise interpretation in terms of probabilities or metabolite flow per unit time. The methodology can be applied both in the absence of a specific biological context, or tailored to different environmental conditions by incorporating flux distributions computed from constraint-based modelling (e.g., Flux-Balance Analysis). I will illustrate the approach on the central carbon metabolism of Escherichia coli, revealing drastic changes in the topological and community structure of the metabolic graphs, which capture the re-routing of metabolic fluxes under each growth condition.

By integrating Flux Balance Analysis and tools from network science, our framework allows for the interrogation of environment-specific metabolic responses beyond fixed, standard pathway descriptions.

  • Industrial and Applied Mathematics Seminar
19 January 2017

Averaging, either spatial or temporal, is a powerful technique in complex multi-scale systems.

However, in some situations it can be difficult to justify.

For example, many real-world networks in technology, engineering and biology have a function and exhibit dynamics that cannot always be adequately reproduced using network models given by the smooth dynamical systems and fixed network topology that typically result from averaging. Motivated by examples from neuroscience and engineering, we describe a model for what we call a "functional asynchronous network". The model allows for changes in network topology through decoupling of nodes and stopping and restarting of nodes, local times, adaptivity and control. Our long-term goal is to obtain an understanding of structure (why the network works) and how function is optimized (through bifurcation).

We describe a prototypical theorem that yields a functional decomposition for a large class of functional asynchronous networks. The result allows us to express the function of a dynamical network in terms of individual nodes and constituent subnetworks.


  • Industrial and Applied Mathematics Seminar
8 December 2016
John Hutchinson

The stability of structures continues to be scientifically fascinating and technically important.  Shell buckling emerged as one of the most challenging nonlinear problems in mechanics more than fifty years ago when it was intensively studied.  It has returned to life with new challenges motivated not only by structural applications but also by developments in the life sciences and in soft materials.  It is not at all uncommon for slightly imperfect thin cylindrical shells under axial compression or spherical shells under external pressure to buckle at 20% of the buckling load of the perfect shell.  A historical overview of shell buckling will be presented followed by a discussion of recent work by the speaker and his collaborators on the buckling of spherical shells.  Experimental and theoretical work will be described with a focus on imperfection-sensitivity and on viewing the phenomena within the larger context of nonlinear stability. 

  • Industrial and Applied Mathematics Seminar
1 December 2016
Jon Chapman, Patrick Farrell

A computational and asymptotic analysis of the solutions of Carrier's problem  is presented. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the bifurcation parameter tends to zero. The method of Kuzmak is then applied to construct asymptotic solutions to the problem. This asymptotic approach explains the bifurcation structure identified numerically, and its predictions of the bifurcation points are in excellent agreement with the numerical results. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag is incorrect.

  • Industrial and Applied Mathematics Seminar
24 November 2016
Athanasios Tsanas

In this talk, I am reflecting on the last 8 extremely enjoyable years I spent in the department (DPhil, OCIAM, 2008-2012, post-doc, WCMB, 2012-2016). My story is a little unusual: coming from an Engineering undergraduate background, spending 8 years in the Maths department, and now moving to a faculty position at the Medical School. However, I think it highlights well the enormous breadth and applicability of mathematics beyond traditional disciplinary boundaries. I will discuss different projects during my time in Oxford, focusing on time-series, signal processing, and statistical machine learning methods, with diverse applications in real-world problems.

  • Industrial and Applied Mathematics Seminar
17 November 2016
Stephen Wilson

In the first part of the talk, I will describe a fluid-dynamical model for a "anti-surfactant" solution (such as salt dissolved in water) whose surface tension is an increasing function of bulk solvent concentration. In particular, I will show that this model is consistent with the standard model for surfactants, and predicts a novel instability for anti-surfactants not present for surfactants. Some further details are given in the recent paper by Conn et al. Phys. Rev. E 93 043121 (2016).


In the second part of the talk, I will formulate and analyse the governing equations for the flow of a thixotropic or antithixotropic fluid in a slowly varying channel. These equations are equivalent to the equations of classical lubrication theory for a Newtonian fluid, but incorporate the evolving microstructure of the fluid, described in terms of a scalar structure parameter. If time permits, I will seek draw some conclusions relevant to thixotropic flow in porous media. Some further details are given in the forthcoming paper by Pritchard et al. to appear in J Non-Newt. Fluid Mech (2016).

  • Industrial and Applied Mathematics Seminar
10 November 2016
OCIAM Group Meeting

Ousman Kodio

Lubricated wrinkles: imposed constraints affect the dynamics of wrinkle coarsening

We investigate the problem of an elastic beam above a thin viscous layer. The beam is subjected to
a fixed end-to-end displacement, which will ultimately cause it to adopt the Euler-buckled
state. However, additional liquid must be drawn in to allow this buckling. In the interim, the beam
forms a wrinkled state with wrinkles coarsening over time. This problem has been studied
experimentally by Vandeparre \textit{et al.~Soft Matter} (2010), who provides a scaling argument
suggesting that the wavelength, $\lambda$, of the wrinkles grows according to $\lambda\sim t^{1/6}$.
However, a more detailed theoretical analysis shows that, in fact, $\lambda\sim(t/\log t)^{1/6}$.
We present numerical results to confirm this and show that this result provides a better account of
previous experiments.


Edward Rolls

Multiscale modelling of polymer dynamics: applications to DNA

We are interested in generalising existing polymer dynamics models which are applicable to DNA into multiscale models. We do this by simulating localized regions of a polymer chain with high spatial and temporal resolution, while using a coarser modelling approach to describe the rest of the polymer chain in order to increase computational speeds. The simulation maintains key macroscale properties for the entire polymer. We study the Rouse model, which describes a polymer chain of beads connected by springs by developing a numerical scheme which considers the a filament with varying spring constants as well as different timesteps to advance the positions of different beads, in order to extend the Rouse model to a multiscale model. This is applied directly to a binding model of a protein to a DNA filament. We will also discuss other polymer models and how it might be possible to introduce multiscale modelling to them.

  • Industrial and Applied Mathematics Seminar
3 November 2016
Henry Schenck

One of the fundamental tools in numerical analysis and PDE
is the finite element method (FEM). A main ingredient in
FEM are splines: piecewise polynomial functions on a
mesh. Even for a fixed mesh in the plane, there are many open
questions about splines: for a triangular mesh T and
smoothness order one, the dimension of the vector space
  C^1_3(T) of splines of polynomial degree at most three
is unknown. In 1973, Gil Strang conjectured a formula
for the dimension of the space C^1_2(T) in terms of the
combinatorics and geometry of the mesh T, and in 1987 Lou
Billera used algebraic topology to prove the conjecture
(and win the Fulkerson prize). I'll describe recent progress
on the study of spline spaces, including a quick and self
contained introduction to some basic but quite useful tools
from topology.

  • Industrial and Applied Mathematics Seminar
27 October 2016

The study of moving contact lines is challenging for various reasons: Physically no sliding motion is allowed with a standard no-slip boundary condition over a solid substrate. Mathematically one has to deal with a free-boundary problem which contains certain singularities at the contact line. Instabilities can lead to topological transition in configurations space - their rigorous mathematical understanding is highly non-trivial. In this talk some state-of-the-art modeling and numerical techniques for such challenges will be presented. These will be applied to flows over solid and liquid substrates, where we perform detailed comparisons with experiments.

  • Industrial and Applied Mathematics Seminar