Monday, 19 March 2018

Knots and surfaces - the fascinating topology of n-manifolds

Oxford Mathematician Andras Juhasz discusses and illustrates his latest research into knot theory.

"We can only see a small part of Space, even with the help of powerful telescopes. This looks like 3-dimensional coordinate space, but globally it might have a more complicated shape. An n-dimensional manifold, or n-manifold in short, is a space that locally looks like the standard n-dimensional coordinate space, whose points we can describe with n real coordinates. Topology considers such spaces up to continuous or smooth deformations, as if they were made out of rubber.

The only connected 1-manifolds are the real line and the circle. 2-dimensional manifolds are also called surfaces. The closed oriented (or 2-sided) surfaces are the sphere, the surface of a doughnut (the torus), or the surface of a doughnut with several holes. The number of holes is called the genus of  the surface, and is an example of a topological invariant: an algebraic object (e.g., a number, polynomial, or vector space) assigned to a space that is unchanged by deformations. We have already seen that we live in a 3-manifold, and, if we add the time dimension, in a 4-dimensional spacetime.


2-manifolds (please view all films in Chrome, Firefox or Explorer):


Genus                 0                                     1                                               2


The theory of 1- and 2-manifolds is classical. Surprisingly, dimensions greater than 4 are simpler than dimensions 3 and 4, due to the fact that there is enough space to perform a certain topological trick that allows one to reduce the classification problem to algebra. The focus of modern topology is hence in dimensions 3 and 4.  While 3-manifold topology is closely related to geometry, the theory of smooth 4-manifolds is more analytical. In dimension 4, the difference between smooth and continuous deformations becomes essential. For example, there is just one 4-manifold that looks like 4-dimensional coordinate space up to continuous deformation, but infinitely many of these are smoothly different.

A knot is a circle embedded in 3-space, up to deformation. (Topologically, a knot on a string is always trivial, as one can just pull one end along the string itself until the knot disappears.) A link is a collections of knots that link with each other (hence the name). These play an important role in low-dimensional topology, since every 3- and 4-manifold can be described by a link whose components are each labelled by an integer.

Deformation of an unknot:





Knot Floer homology is an invariant of links defined independently by Ozsváth-Szabó and Rasmussen in 2002. It assigns a finite-dimensional vector space to every link, and contains important geometric information.

Two links are cobordant if they can be connected by a surface in 4-space. If we think of the fourth coordinate as time, each time slice gives a (possibly singular) link. As time varies from say 0 to 1, we obtain a movie of links. In a recent paper published in Advances in Mathematics, I have shown that a link cobordism induces a linear map on knot Floer homology. This can be used to understand the possible surfaces links can bound in 4-space, which is closely related to the topology of smooth 4-manifolds".

A link cobordism:





Thursday, 15 March 2018

Understanding plasma-liquid interactions

Oxford Mathematician John Allen, Professor Emeritus of Engineering Science, talks about his work on the electrohydrodynamic stability of a plasma-liquid interface. His collaborators are Joshua Holgate and Michael Coppins at Imperial College.

 '"The study of plasma-liquid interactions is an increasingly important topic in the field of plasma science and technology with applications in nanoparticle synthesis, catalysis of chemical reactions, material processing, water treatment, sterilization and plasma medicine. This particular work is motivated by the plasma-liquid interactions inherent in magnetic confinement fusion devices, such as tokamaks, either due to melt damage of the metal walls or in new liquid metal divertor concepts. The ejection of molten droplets has been observed in both cases and is of considerable concern to the operation of a successful fusion device. Understanding the stability of the liquid metal surface is a critical issue.

Previously-studied instabilities of liquid metal surfaces in tokamaks include a Kelvin-Helmholtz instability due to plasma flow across the metal surface, a Rayleigh-Taylor instability driven by the j × B force due to a current flowing in the metal, a Rayleigh-Plateau instability of the liquid metal rim around a cathode arc spot crater, and droplet emission from bursting bubbles which are formed by liquid boiling or absorption of gases from the plasma. However, none of these studies considers the effect of the strong electric fields and ion flows in the sheath region between the plasma and the liquid surface despite the observations of electrical effects such as arcing, which cause considerable damage to the tokamak wall, and enhanced droplet emission rates from electrically-biased surfaces. Furthermore electrostatic breakup has been identified as an important process for liquid droplets in plasmas.

Instabilities driven by electric fields, i.e. electrohydrodynamic (EHD) instabilities, at the interface between a conducting liquid and vacuum, were originally studied by Melcher and subsequently by Taylor and McEwan. Melcher’s marginal stability criterion was invoked by Bruggeman et al. in order to explain the filamentary structure of a glow discharge over a water cathode and, additionally, to explain the instability of an electrolytic water solution cathode from an earlier experiment. Earlier evidence for EHD instabilities of the plasma-liquid interface appears in an experiment on unrelated work where an arc spot occasionally formed on an electrically-isolated mercury pool which was in contact with the plasma. Another EHD effect, the deformation of a liquid surface into a Taylor cone, has recently been used to form the cathode of a corona discharge.

Our work investigates the EHD stability of a plasma-liquid interface with a linear perturbation analysis. Melcher’s stability criterion is found to apply to short-wavelength perturbations of the surface. However the fast-moving ions in the sheath provide a significant pressure on the liquid surface which can overcome the electric stress for long-wavelength perturbations. This effect has been neglected in previous studies and provides an overall increase in the critical voltage which must be applied to the surface in order to make it unstable. This effect is encouraging for the ongoing development of new plasma-liquid technologies."

Thursday, 15 March 2018

John Ball wins Leonardo da Vinci Award

Oxford Mathematician John Ball has won the European Academy of Sciences Leonardo da Vinci award. The award is given annually for outstanding lifetime scientific achievement. In the words of the Committee,  "through a research career spanning more than 45 years, Professor Ball has made groundbreaking and highly significant contributions to the mathematical theory of elasticity, the calculus of variations, and the mathematical analysis of infinite-dimensional dynamical systems."

Sir John Ball FRS is Sedleian Professor of Natural Philosophy in the University of Oxford and Director of the Oxford Centre for Nonlinear Partial Differential Equations. He is a Fellow of The Queen's College.



Monday, 12 March 2018

Oxford Summer School on Economic Networks 25-29 June 2018 - register by 15 March

The Oxford Summer School on Economic Networks, hosted by Oxford Mathematics and the Institute of New Economic Thinking, aims to bring together graduate students from a range of disciplines (maths, statistics, economics, policy, geography, development, ..) to learn about the techniques, applications and impact of network theory in economics and development. 
We look forward to welcoming a large number of world renowned experts in economic networks and complexity science. Confirmed speakers for the 2018 edition include Fernando Vega-Redondo, Mihaela van der Schaar, Rama Cont, Doyne Farmer, Pete Grindrod, Renaud Lambiotte, Elsa Arcaute and Taha Yasseri. Tutorials and lectures include social networks, games and learning, financial networks, economic complexity and urban systems.
Alongside a rigorous academic schedule, the summer school also includes a walking tour of the historic university and city centre, a punting trip on the river Cherwell and a dinner in one of Oxford's historic colleges.
The deadline for applications is March 15th - more information is available here. Please contact us at with any questions.

Friday, 9 March 2018

How do airlines gauge unknown demand?

Oxford Mathematicians Ilan Price and Jaroslav Fowkes discuss their work on unconstraining demand with Gaussian Processes.

"One of the key revenue management challenges which airlines, hotels, cruise ships (and other industries) all share is the need to make business decisions in the face of constrained (or censored) demand data.

Airlines, for example, commonly set booking limits on the number of cheaper fare-classes that can be purchased, or make cheaper fare-classes unavailable for booking at certain times, in an attempt to divert some of that demand to the more expensive tickets still available. While a fare-class on a given flight route is available for booking, the demand for that 'product', at that price, is accurately captured by its total recorded bookings. However, once the product has been unavailable for booking for a period of time, recorded bookings no longer capture true demand, and the demand data is said to be 'constrained' or 'censored'.

Practices which constrain demand data pose a big challenge for successful revenue management. This is because many important decisions, including setting ticket prices, making changes to an airline's flight network, adding or removing capacity on a certain route, and many others, are all heavily dependent on accurate historical demand data. Moreover, precisely those decisions regarding which fare-classes to make unavailable (and for what periods of time) themselves depend on accurate demand data. Thus predicting what demand would have been had it not been constrained - known as 'unconstraining demand' - is an important research problem.

Our research proposes a new approach to this problem, using a model developed within the framework of Gaussian process (GP) regression. The general idea behind Gaussian process regression is very intuitive: we start by assuming a prior Gaussian distribution over functions, and then we condition that distribution on the observed data, so as to restrict the set of likely functions to only those functions which make sense given the observed data. More precisely, our goal is to infer the posterior predictive distribution $p(f^* | y, X, X^*)$, where $f^*$ are the values of the function evaluated at some prediction points $X^*$, and $y$ are the observed data at points $X$. We can then use the mean of this distribution as our predictions. 

Figure 1: Illustration of GP regression for unconstraining demand. The figure on the left shows the mean prediction and confidence interval produced by our GP method, based on the true demand observations. The dotted black line indicates when the booking limit was reached, and the red line beyond this point shows the GP's unconstrained approximations. The figure on the right shows in red the reconstruction of the cumulative demand curve over the constrained period using the daily demand values predicted with the GP.

In the course of this inference procedure, we need to specify (i) a likelihood function or observation model, and (ii) a mean and covariance function for the GP prior.

Our model uses a Poisson likelihood, based on our implicit model of the bookings process as a doubly stochastic Poisson process, i.e. where bookings are determined by a Poisson process whose rate $\lambda$ is itself a Gaussian process (and thus changes over time).

For the GP prior, we use a zero mean function and define a new 'variable degree polynomial covariance function' \begin{equation} k(x,x') = \sigma^2(x^\top x' + c)^p, \end{equation} with $\theta_c = \{\sigma , c, p\}$ as the covariance hyperparameters (a modification of the polynomial covariance function in which $p$ is a fixed positive integer).

Having conducted a number of numerical experiments, our results are rather promising: the method compares favourably with state of the art methods when repeating experiments from recent literature. The added benefit, though, is that when these experiments are modified to have weaker assumptions on how the test data should look and be generated, our method maintains its strong performance better than its competitors. Our modifications included diversifying the shape of demand curve on which the methods were tested, as well as allowing for the presence of changepoints - points at which the characteristics of the underlying demand trend change dramatically. Using existing theory, we can elegantly extend our GP regression framework to cope with such situations by constructing an appropriate covariance function. For our purposes, we want to allow for the fact that the covariance before and after the changepoint might be completely different. We therefore redefine our covariance function to be \begin{align}\label{eq: Changepoint covariance} k(x,x') = \begin{cases} \sigma_1^2(x^\top x' + c_1)^{p_1} & \text{if } x,x' < x_c,\\ \sigma_2^2(x^\top x' + c_2)^{p_2} & \text{if } x,x' \geq x_c,\\ 0 & \text{otherwise}, \end{cases} \end{align} where $\theta = \{\sigma_1, \sigma_2, c_1, c_2, p_1, p_2, x_c\}$ are all hyperparameters inferred from the data. You can see an example of how well it performs in the image below."

 Figure 2: Illustration of automatic changepoint detection with GPs using our piecewise-defined variable degree polynomial covariance function.

You can read the research in greater detail here.

Monday, 5 March 2018

The brains of the matter. Understanding the cerebral cortex

The brain is the most complicated organ of any animal, formed and sculpted over 500 million years of evolution. And the cerebral cortex is a critical component. This folded grey matter forms the outside of the brain, and is the seat of higher cognitive functions such as language, episodic memory and voluntary movement.

The cerebral cortex of mammals has a unique layered structure where different types of neuron reside. The thickness of the cortical layer is roughly the same across different species, while the cortical surface area shows a dramatic increase (1000 fold from mouse to human). This difference underlies a significant expansion in the number of cortical neurons produced in the course of embryonic development, resulting in the increased function and complexity of the adult brain. A human cortex accommodates 16 billion neurons as opposed to a mouse’s mere 14 million.

Key elements of this problem are being addressed by Oxford Mathematical Biologist Noemi Picco in a new collaboration involving an interdisciplinary team of mathematicians including Philip Maini in Oxford and Thomas Woolley in  Cardiff and biologists Zoltán Molnár from the Department of Physiology, Anatomy and Genetics in Oxford and Fernando García-Moreno at the Achucarro Basque Center for Neuroscience in Bilbao.

In particular the team are developing a mathematical model of cortical neurogenesis, the process by which neurons grow and develop in the cerebral cortex. Given that species diversity originates from the divergence of developmental programmes, understanding the cellular and molecular mechanisms regulating cell number and diversity is critical for shedding light on cortex evolution.

Many factors influence how neurogenesis in the cortex differs between species, including the types of neurons and neural progenitor cells, the different ways in which they proliferate and differentiate, and the length of the process (85 days in a human, 8 days in a mouse). This project combines mathematical modelling and experimental observations to incorporate these different factors. A key determinant of the neuronal production is the modulation of proliferative (self-amplifying) and differentiative (neurogenic) divisions. By modelling the temporal changes in the propensity of different cell division types, we are able to identify the developmental programme that can justify the observed number of neurons in the cortex.

The growing availability of species-specific experimental data will allow the researchers to map all the possible evolutionary pathways of the cortex, and create a mathematical framework that is general enough to encompass all cortex developmental programmes, while being specifiable enough to be descriptive of single species. This, in turn, has the potential to create a new way to identify developmental brain disorders as deviations from the normal developmental program, giving a mechanistic insight into their cause and clinically actionable suggestions to correct them.

As part of the project, Noemi has released a Neurogenesis Simulator, an app that allows experimentalists to ‘play’ with the mathematical model, choosing the species and the model and calibrating the parameters, to observe how the model outcome changes without having to worry about the mathematical formulation and thereby generating even further cross-disciplinary collaboration.

Noemi’s work is supported by St John’s College Research Centre. Click here for the published article.

Monday, 5 March 2018

Euler, the Secrets of Applied Mathematics and Inverse Problems - our latest round-up of books by Oxford Mathematicians

'Euler's Pioneering Equation' has been compared to a Shakespearean Sonnet. But even if you don't buy that, Robin Wilson's book does much to show how an 18th century Swiss mathematician managed to bring together the five key constants in the subject: the number 1, the basis of our counting system; the concept of zero, which was a major development in mathematics, and opened up the idea of negative numbers; π an irrational number, the basis for the measurement of circles; the exponential e, associated with exponential growth and logarithms; and the imaginary number i, the square root of -1, the basis of complex numbers. Some achievement.

We are always being told that mathematics impacts every corner of our lives -  our security, our climate, even our very selves. Want a quick summary of how? Alain Goriely's Applied Mathematics: A Very Short Introduction does just that, laying out the basics of the subject and exploring its range and potential. If you want to know how cooking a turkey and medical imaging are best explained by mathematics (or even if you don't) this is an excellent read.

By contrast Yves Capdeboscq together with colleague Giovanni S. Alberti from Genoa has published 'Lectures on Elliptic Methods For Hybrid Inverse Problems based on a series of 2014 lectures. Targeting the Graduate audience, this work tackles one of the most important aspects of the mathematical sciences: the Inverse Problem. In the words of the authors "Inverse problems correspond to the opposite (of a direct problem), namely to find the cause which generated the observed, measured result." 

Click here for our last literary selection including Prime Numbers, Networks and Russian Mathematicians.

Friday, 2 March 2018

Oxford Mathematician Robin Wilson awarded the 2017 Stanton Medal

Oxford Mathematician Robin Wilson has been awarded the 2017 Stanton Medal. The medal is awarded every two years by the Institute of Combinatorics and its Applications (ICA) for outreach activities in combinatorial mathematics.

In the words of the ICA citation, "Robin Wilson has, for fifty years, been an outstanding ambassador for graph theory to the general public.  He has lectured widely (giving some 1500 public lectures), and extended the reach of his lectures through television, radio, and videotape.  He has also published extensively on combinatorial ideas, written in a style that is engaging and accessible.  He has provided direction, encouragement, and support to colleagues and students at all levels. His superb talents at conveying the beauty of graph-theoretic ideas, and inviting his readers and listeners to join in, have enthused many students, teachers, and researchers. Professor Wilson’s advocacy and outreach for combinatorics continue to yield many positive impacts that are enjoyed by researchers and non-specialists alike."

Robin Wilson is an Emeritus Professor of Pure Mathematics at the Open University, Emeritus Professor of Geometry at Gresham College, London, and a former Fellow of Keble College, Oxford. He is the author of many books including 'Combinatorics: A Very Short Introduction', 'Four Colours Suffice: How the Map Problem Was Solved,' 'Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life' and his textbook ‘Introduction to Graph Theory.’ His latest Oxford Mathematics Public Lecture on Euler's pioneering equation can be watched here.


Monday, 26 February 2018

Euler's beautiful brain and everyone else's - Oxford Mathematics Public Lectures

We have two contrasting Oxford Mathematics Public Lectures coming up in the next ten days. One features a genius from the eighteenth century whose work is still pertinent today. The other is very much from the 21st century and illuminates the direction mathematics is currently travelling. Please email to register or follow our twitter account for details on how to watch live.

Euler’s pioneering equation: ‘the most beautiful theorem in mathematics’ - Robin Wilson. 28 February, 2018, 5-6pm

Can Mathematics Understand the Brain? - Alain Goriely, March 8th, 5.15-6.15pm


Euler’s pioneering equation: ‘the most beautiful theorem in mathematics’ - Robin Wilson

Euler’s equation, the ‘most beautiful equation in mathematics’, startlingly connects the five most important constants in the subject: 1, 0, π, e and i. Central to both mathematics and physics, it has also featured in a criminal court case and on a postage stamp, and has appeared twice in The Simpsons. So what is this equation – and why is it pioneering?

Robin Wilson is an Emeritus Professor of Pure Mathematics at the Open University, Emeritus Professor of Geometry at Gresham College, London, and a former Fellow of Keble College, Oxford.

28 February 2018, 5pm-6pm, Mathematical Institute, Oxford. Please email to register

Can Mathematics Understand the Brain? - Alain Goriely

The human brain is the object of the ultimate intellectual egocentrism. It is also a source of endless scientific problems and an organ of such complexity that it is not clear that a mathematical approach is even possible, despite many attempts. 

In this talk Alain will use the brain to showcase how applied mathematics thrives on such challenges. Through mathematical modelling, we will see how we can gain insight into how the brain acquires its convoluted shape and what happens during trauma. We will also consider the dramatic but fascinating progression of neuro-degenerative diseases, and, eventually, hope to learn a bit about who we are before it is too late. 

Alain Goriely is Professor of Mathematical Modelling, University of Oxford and author of 'Applied Mathematics: A Very Short Introduction.'

8 March, 5.15 pm-6.15pm, Mathematical Institute, Oxford. Please email to register


Monday, 19 February 2018

Combinatorics - past, present and future

Oxford Mathematician Katherine Staden provides a fascinating snapshot of the field of combinatorics, and in particular extremal combinatorics, and the progress that she and her collaborators are making in answering one of its central questions posed by Paul Erdős over sixty years ago. 

"Combinatorics is the study of combinatorial structures such as graphs (also called networks), set systems and permutations. A graph is an encoding of relations between objects, so many practical problems can be represented in graph theoretic terms; graphs and their mathematical properties have therefore been very useful in the sciences, linguistics and sociology. But mathematicians are generally concerned with theoretical questions about graphs, which are fascinating objects for their own sake. One of the attractions of combinatorics is the fact that many of its central problems have simple and elegant formulations, requiring only a few basic definitions to be understood. In contrast, the solutions to these problems can require deep insight and the development of novel tools.

A graph $G$ is a collection $V$ of vertices together with a collection $E$ of edges. An edge consists of two vertices. We can represent $G$ graphically by drawing the vertices as points in the plane and drawing a (straight) line between vertices $x$ and $y$ if $x,y$ is an edge.

Extremal graph theory concerns itself with how big or small a graph can be, given that it satisfies certain restrictions. Perhaps the first theorem in this area is due to W. Mantel from 1907, concerning triangles in graphs. A triangle is what you expect it to be: three vertices $x,y,z$ such that every pair $x,y$ and $y,z$ and $z,x$ is an edge. Consider a graph which has some number $n$ of vertices, and these are split into two sets $A$ and $B$ of size $\lfloor n/2\rfloor$, $\lceil n/2\rceil$ respectively. Now add every edge with one vertex in $A$ and one vertex in $B$. This graph, which we call $T_2(n)$, has $|A||B|=\lfloor n^2/4\rfloor$ edges. Also, it does not contain any triangles, because at least two of its vertices would have to both lie in $A$ or in $B$, and there is no edge between such pairs. Mantel proved that if any graph other than $T_2(n)$ has $n$ vertices and at least $\lfloor n^2/4\rfloor$ edges, it must contain a triangle. In other words, $T_2(n)$ is the unique `largest' triangle-free graph on $n$ vertices.

Following generalisations by P. Turán and H. Rademacher in the 1940s, Hungarian mathematician Paul Erdős thought about quantitatively extending Mantel's theorem in the 1950s. He asked the following: among all graphs with $n$ vertices and some number $e$ of edges, which one has the fewest triangles? Call this quantity $t(n,e)$. (One can also think about graphs with the most triangles, but this turns out to be less interesting).

Astoundingly, this seemingly simple question has yet to be fully resolved, 60 years later. Still, in every intervening decade, progress has been made, by Erdős, Goodman, Moon-Moser, Nordhaus-Stewart,  Bollobás, Lovász-Simonovits, Fisher and others. Finally, in 2008, Russian mathematician A. Razborov managed to solve the problem asymptotically (meaning to find an approximation $g(e/\binom{n}{2})$ to $t(n,e)$ which is arbitrarily accurate as $n$ gets larger). Razborov showed that, for large $n$, $g(e/\binom{n}{2})$ has a scalloped shape: it is concave between the special points $\frac{1}{2}\binom{n}{2}, \frac{2}{3}\binom{n}{2}, \frac{3}{4}\binom{n}{2}, \ldots$. His solution required him to develop the new method of flag algebras, part of the emerging area of graph limits, which has led to the solution of many longstanding problems in extremal combinatorics.

The remaining piece of the puzzle was to obtain an exact (rather than asymptotic) solution. In recent work with Hong Liu and Oleg Pikhurko at the University of Warwick, I addressed a conjecture of Lovász and Simonovits, the solution of which would answer Erdős's question in a strong form. The conjecture put forward a certain family of $n$-vertex, $e$-edge graphs which are extremal, in the sense that they should each contain the fewest triangles. So in general there is more than one such graph, one aspect which makes the problem hard. Building on ideas of Razborov and Pikhurko-Razborov, we were able to solve the conjecture whenever $e/\binom{n}{2}$ is bounded away from $1$; in other words, as long as $e$ is not too close to its maximum possible value $\binom{n}{2}$.

Our proof spans almost 100 pages and (in contrast to Razborov's analytic proof) is combinatorial in nature, involving a type of stability argument. It would be extremely interesting to close the gap left by our work and thus fully answer Erdős's question."

Revolving captions:

The graph $T_2(n)$, which is the unique largest triangle-free graph on $n$ vertices.

The minimum number of triangles $t(n,e)$ in an $n$-vertex $e$-edge graph plotted against $e/\binom{n}{2}$. This was proved in the pioneering work of A. Razborov.

Making new graphs from old: an illustration of a step in the proof of the exact result by Liu-Pikhurko-Staden.