Monday, 6 February 2017

Oxford Mathematics Research - On the null origin of the ambitwistor string

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Eduardo Casali discusses his work.

This work [1], done with my collaborator Piotr Tourkine, is our attempt to understand the origin of some recent constructions in theoretical physics and how they fit into more standard techniques. These constructions go by the name of ambitwistor strings [2], named so because they combine elements of classical twistor theory and string theory. As in usual string theory, it is a theory of maps from Riemann surfaces into some target space, usually spacetime, taken to be $\mathbb{R}^D$ where the dimension $D$ depends on which flavour of string theory one is studying. In the case of the ambitwistor string target space is a generalization of twistor space to arbitrary dimensions called ambitwistor space. This can be defined as the space of complex null geodesics of complexified spacetime. For example, the ambitwistor space of $\mathcal{M}=\mathbb{C}^D$ is given by the quotient $\mathbb{A}=T^*\mathcal{M}//\{P^2=0\}$. The ambitwistor string is then a theory of holomorphic maps from Riemann surfaces into $\mathbb{A}$.

Following the rules of string theory, one can allow certain kinds of singularities on the Riemann surface such that these surfaces now have moduli and the string theory gives an integral over the corresponding moduli space. These integrals correspond to scattering of particles in a theory with an infinite spectrum of massive particles. To obtain results related to standard quantum field theories a low-energy limit must be taken, which corresponds to $\alpha'\rightarrow0$ where $\alpha'$ is the inverse of the string tension. In terms of the moduli space integral this limit is subtle, breaking the integral into several smaller pieces akin to Feynman diagrams. But surprisingly, in the ambitwistor string no limit needs to be taken. The moduli space integral already gives the result in the $\alpha'\rightarrow0$ limit. This is possible since the integral in this case localizes to the solution set of a set of equations called the scattering equations[3]. These equations had previously been found in the opposite limit of the string, the high-energy limit $\alpha'\rightarrow\infty$[4], so their appearance in the ambitwistor string is a bit of a puzzle. Another related puzzle was how the ambitwistor string fits into the framework of conventional string theory. It shares several similarities, both in its set-up and in the calculations coming from it, but a naive attempt to take the $\alpha'\rightarrow0$ limit gives a very different result.

It is here that our recent work comes in. We showed how the ambitwistor string model can be derived from a more fundamental model, the null string. This third string theory is obtained by taking the $\alpha'\rightarrow\infty$ limit in the original string and then quantising. By making specific choices in the null string one can show that it coincides with the ambitwistor string. With this interpretation of the ambitwistor string in hand we can make sense of some of its more puzzling characteristics. First, it gives a rationale for the appearance of the scattering equations in what seemed to be the complete opposite limit. It also helps connecting the ambitwistor string to usual string theory and shows promise in generalizing the ambitwistor string to describe more general theories. But more importantly, it opens a new way in which we can approach it. By making different choices when setting up the null string, which don't affect the end result, we hope to obtain new scattering formulas and a new understanding of the geometric role played by the scattering equations and the moduli space of Riemann surfaces in these formulas. Finally, this might also shed light on the old problem of how the spacetime equations of motions are codified into ambitwistor space. An answer to this was given by LeBrun and Mason [5], but their construction is quite different from what the ambitwistor string seems to imply. That is, that the equations of motion should be somehow codified into embeddings of Riemann surfaces into ambitwistor space.

[1] E. Casali and P. Tourkine, “On the null origin of the ambitwistor string,” JHEP 11 (2016).

[2] L. Mason and D. Skinner, “Ambitwistor strings and the scattering equations,” JHEP 07 (2014). 

[3] F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles in Arbitrary Dimensions,” Phys. Rev. Lett. 113 no. 17, (2014).

[4] D. J. Gross and P. F. Mende, “The High-Energy Behavior of String Scattering Amplitudes,” Phys. Lett. B197 (1987) 129–134.

[5] C. LeBrun, “Spaces of complex null geodesics in complex-Riemannian geometry,” Trans. Amer. Math. Soc. 278 no. 1, (1983) 209–231.

Saturday, 4 February 2017

Oxford Mathematics Research - Rates of convergence in the method of alternating projections

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician David Seifert discusses his and his collaborator Catalin Badea's work.

Given a point $x$ and a shape $M$ in three-dimensional space, how might we find the point in $M$ which is closest to $x$? In general there need not be an easy answer, but suppose we have the extra information that $M$ is in fact the intersection of several sets which are much simpler to handle (so that a point lies in $M$ if and only if it lies in each of the simpler sets). In this case we might put the sets in order and proceed iteratively, by letting $x=x_0$ be the starting point in a sequence and taking $x_1$ to be the point closest to $x_0$ among the points in the first of the simpler sets. Next we find the point $x_2$ in the second simple subset which lies closest to $x_1$, and we continue in this way, returning to the first simple subset when we have exhausted the list. Does this process lead to the answer we want?

Recent research by Oxford Mathematician David Seifert and his collaborator Catalin Badea of Université de Lille 1 tackles this question not just in familiar three-dimensional space but in the much more general setting of Hilbert spaces. (Here and in what follows it is more the broad underlying ideas that matter, not so much the mathematical details.) Given a Hilbert space $X$ and a closed subspace $M$ of $X$, the orthogonal projection $P_M$ onto $M$ is the linear operator defined by the property that $P_M(x)$ is the point in $M$ which lies closest to $x\in X$. Figure 1 illustrates the case where $X$ is the Euclidean plane and $M$ is a line through the origin, but $X$ could also be a Sobolev space or some other infinite-dimensional Hilbert space. Many problems in mathematics, from linear algebra to the theory of PDEs, involve finding $P_M(x)$ for a given vector $x\in X$ and a space $M$. Sometimes $P_M(x)$ can be computed easily, but in other cases it cannot. Nevertheless, there is a natural way of finding an approximate solution if it is possible, as in the initial example, to break down our problem into easier subproblems.

Indeed, suppose that $M=M_1\cap\dotsc\cap M_N$ where $M_1,\dots, M_N$ are themselves closed subspaces of $X$. We may not know much about $M$ or the operator $P_M$, but often the problem of finding the nearest point to a given vector $x\in X$ in any of the subspaces $M_k$, $1\le k\le N$, is much simpler. Writing $P_k$ for the orthogonal projection onto $M_k$, $1\le k\le N$, we may then successively find the vectors $P_1(x)$, $P_2P_1(x), \dots, P_N\cdots P_1(x)$, $P_1P_N\cdots P_1(x)$ and so on, projecting cyclically onto the subspaces $M_1,\dots,M_N$; see Figure 2.


This method of alternating projections has many different applications. These include surprising ones such as image restoration and computed tomography, but also linear algebra and the theory of PDEs. In linear algebra it corresponds to solving a system of linear equations one by one, at each stage finding the solution to the next equation which lies closest to the previous solution (the Kaczmarz method); in the theory of PDEs the method can capture the process of solving an elliptic PDE on a composite domain by solving it cyclically on each subdomain and using the boundary conditions to update the solution at each stage (the Schwarz alternating method). In general one is led to consider the single operator $T=P_N\cdots P_1$. It is known [2] that

$$ \|T^n(x)-P_M(x)\|\to0,\quad n\to\infty, \quad\quad\quad\quad(*) $$

for all $x\in X$, which means that by projecting cyclically onto the subspaces $M_1,\dotsc,M_N$ we may approximate the unknown solution $P_M(x)$ to arbitrary precision. In practice, though, this result is of limited value unless one has some knowledge of the rate at which the convergence takes place in $(*)$, so that one can estimate the number of iterations required to guarantee a specified level of precision. For example, in the Schwarz alternating method should we expect to require 50 iterations or 50,000 iterations in order to achieve a reasonably good approximation to the (unknown) true solution of our PDE?

There is a surprising dichotomy for the rate of convergence in $(*)$. Either the convergence is exponentially fast for all initial vectors $x\in X$, or one can make the convergence as slow as one likes by choosing appropriate initial vectors $x\in X$. In the example in Figure 2 the rate of convergence is determined by the angle between the lines $M_1$, $M_2$, and likewise in the general case the rate of convergence in $(*)$ depends in an interesting way on the geometric relationship between the subspaces $M_1,\dots,M_N$. In the case of the Schwarz alternating method, for instance, the crucial factor is the precise way in which the different subdomains overlap. If they overlap nicely then we will get exponentially fast convergence no matter where we start, and 50 iterations may well be enough to guarantee a good degree of approximation to the true solution. On the other hand, if the domains overlap in an unfavourable way then for certain starting points even 50,000 iterations may be insufficient. So is all lost if one is in the bad case of arbitrarily slow convergence?

Badea and Seifert showed in [1] that the answer is 'no'. More precisely they proved that even in the bad case there exists a dense subspace $X_0$ of $X$ such that for initial vectors $x\in X_0$ the rate of convergence in $(*)$ is faster than $n^{-k}$ for all $k\ge1$. This result provides a theoretical justification for a phenomenon observed by some practitioners, namely that even in bad cases one can usually achieve reasonably rapid convergence without having to experiment with too many different initial vectors $x\in X$. Badea and Seifert succeeded in improving the known results in other ways, too, for instance by giving a sharper estimate on the precise rate of convergence in the case where it is exponential. Underlying these results is the theory of (unconditional) Ritt operators. The theory of Ritt operators is closely related to some of David's earlier work [3, 4] on the quantified asymptotic behaviour of operators, and it is through this connection with operator theory that he first became interested in the method of alternating projections.

Is this the end of the story? As usual in mathematical research, answering one question opens up many more. In particular, Badea and Seifert's main result shows only that there is, in some sense, a rich supply of initial vectors leading to a decent rate of convergence in the method of alternating projections, but it does not say where these vectors should lie. This is an important open problem in approximation theory, and part of David Seifert's current research is concerned with developing techniques which can shed light on questions such as this.


  1. C. Badea and D. Seifert. Ritt operators and convergence in the method of alternating projections. J. Approx. Theory, 205:133–148, 2016.
  2. I. Halperin. The product of projection operators. Acta Sci. Math. (Szeged), 23:96–99, 1962.
  3. D. Seifert. A quantified Tauberian theorem for sequences. Studia Math., 227(2):183–192, 2015.
  4. D. Seifert. Rates of decay in the classical Katznelson-Tzafriri theorem. J. Anal. Math., 130(1):329–354, 2016
Friday, 3 February 2017

Mathematics and health promotion - discussing diabetes on Twitter

Social media for health promotion is a fast-moving, complex environment, teeming with messages and interactions among a diversity of users. In order to better understand this landscape a team of mathematicians and medical anthropologists from Oxford, Imperial College and Sinnia led by Oxford Mathematician Mariano Beguerisse studied a collection of 2.5 million tweets that contain the term "diabetes". In particular, the research focused on two main questions:

(1) Who are the most influential Twitter users that have posted about diabetes?

(2) What themes arise in these tweets?

The researchers used a mixed-methods approach to answer these questions, that relies on techniques from network science, information retrieval, and medical anthropology.

To answer question (1) the team constructed temporal retweet networks, in which the nodes are twitter users, and connections between them exist whenever a user "retweets" a message posted by another. The crucial feature of these networks is that the connections are "directed", that is, there is a distinction between who the author of the tweet is and who retweeted it. The directionality of connections is what allow us to extract the "hub" and "authority" centrality scores for each user in time. In networks, a centrality score is a proxy for importance; hubs and authority scores are useful to distinguish the different roles played by nodes in retweet networks. A good hub is a user that consistently retweets quality tweets, and a good authority is a user who posts them. Whereas the hub landscape is diffuse and has few consistent players, top authorities are highly persistent across time and comprise bloggers, advocacy groups and NGOs related to diabetes, as well as for-profit entities without specific diabetes expertise.

To get a closer look at who the most influential accounts are, the researchers constructed the follower network of the top authorities  (i.e., who follows whom among top authority nodes). An analysis of this network's communities places these top hubs in different groups with a distinct character such as Twitter accounts that are mostly focused on diabetes activism, health and science, lifestyle, commercial accounts, and comedians and parody accounts. 

To answer question (2) the team separated the tweets by weeks, and obtained the topics in each weekly bin using a technique known as "Latent Dirichlet Allocation", which estimates the probability that a tweet containing a specific word belongs to a topic. Once the topics were obtained, the researchers used thematic coding, a technique used by social scientists, to classify them in four broad thematic groups: health information, news, social interaction and commercial. Interestingly, humorous messages and references to popular culture appear consistently more than any other type of tweet. The abundance of jokes about diabetes in online social media is a signal that there is a baseline understanding about the disease and its causes, which may be the result of nutritional heath promotion over the past decades. This observation is at odds with the belief that more health education is required to help people to understand the sorts of foods which might contribute to the development of diabetes.

The results of this work indicate that the diabetes landscape on Twitter is complex, and it cannot be assumed that people can easily discern "good" and "bad" information, and that clearly there is more information available to consumers than they can be expected to absorb. Public health approaches that simply aim to "inform" the public might be insufficient or even be counterproductive, as they make a complicated cacophony of messages even busier. For example, information from bloggers, companies or automated accounts may be in line with broad health recommendations (and indeed may provide a valuable service to users), but without clear distinction from "legitimate" health advice, such information might also push an agenda that could lead to harm or greater health costs in future. In this case, public health agencies may have to develop new approaches to ensure that the electronic health information landscape is one that promotes healthy citizens and not only sweet profits.

Friday, 3 February 2017

Modelling the impact of scientific collaboration

If nations are to grow, both economically and intellectually, they must foster scientific creativity. To do that they must create scientific environments that stimulate collaboration. This is especially true of developing countries as they seek to prosper in a global economy.

Oxford Mathematician Soumya Banerjee’s work looks at scientific collaboration networks, finding novel patterns and clusters in the data that may give insights and guidelines into how the scientific development of developing countries can create richer and more prosperous societies.  

Scientific collaboration networks are an important component of scientific output. Examining a dataset from a scientific collaboration network, Soumya analysed this data using a combination of machine learning techniques and dynamical models.

Soumya's results found a range of clusters of countries with different characteristics of collaboration and corresponding to nations at different stages of development (see figure). Some of these clusters were dominated by developed countries (e.g. the USA and the UK) that have higher numbers of self-connections compared with connections to other countries. Another cluster was dominated by developing nations (such as Liberia and El Salvador) that have mostly connections and collaborations with other countries, but fewer self-connections (shown by different clusters in the figure). 

The research has implications for policy. Countries like El Salvador have a low percentage of foreign connections (this could be a result of the protracted civil war). Consequently the development of active science and research programs in such nations is crucial in generating the concomitant foreign connections. By contrast, Liberia has 100% external connections, suggesting that more effort needs to be taken to develop its own scientific infrastructure. Both a thriving internal and external network are crucial to development.

Proposing a complex systems dynamical model that explains these characteristics, the research explains how the scientific collaboration networks of impoverished and developing nations change over time. The models suggest that developing nations can over time become as successful as the developed nations of today. Soumya also found interesting patterns in the behaviour of countries that may reflect past foreign policies and relations and contemporary geopolitics.

Clearly the model and analyses give food for thought as to how the scientific growth of developing countries can be guided and how it cannot be separated from their existing socio-economic environment and their future prosperity. Big data, machine learning and complexity science are enabling unprecedented computational power to be brought to bear on the fundamental developmental challenges facing humanity.

The figure above plots the percentage of external connections that each country has vs. the distinct number of countries each country is connected with. Clustering is done with k-means and shows three distinct clusters. Click on the image to enlarge.

Soumya's talk on his work can be found here together with his slides and code.


Monday, 30 January 2017

I is for Inverse Problems - the Oxford Mathematics Alphabet

All mathematical models require information to make their predictions; to get something out, you have to put something  in. To predict how an earthquake propagates through the ground, you have to know the material properties of the subsurface rocks. To predict the weather at noon, you have to give the initial conditions at dawn. To predict the drag coefficient of an aircraft, you have to specify its shape.

In many cases, however, we are faced with the opposite problem: given information about the outcome of a physical process, how did it come about? Such a problem is called an inverse problem, in contrast to the forward problems given above, for it inverts the relationship between cause and effect encoded in the underlying equations. Find out more in the latest in our Oxford Mathematics Alphabet.

Monday, 30 January 2017

Statistics: Why the Truth Matters - Tim Harford Public Lecture live podcast details

In our latest Public Lecture Tim Harford, Financial Times columnist and presenter of Radio 4's "More or Less", argues that politicians, businesses and even charities have been poisoning the value of statistics and data. Tim will argue that we need to defend the value of good data in public discourse, and will suggest how to lead the defence of statistical truth-telling.

For details and notification of the live podcast on 8 February at 4pm please click here.



Wednesday, 25 January 2017

Improving the performance of solar cells

Organometal halide perovskite (OMHP) is hardly a household name, but this new material is the source of much interest, not least for Oxford Applied Mathematicians Victor Burlakov and Alain Goriely as they model the fabrication and operation of solar cells.

One of the main advantages of OMHP is that the thin films on its base can be produced in a very inexpensive way via a solution deposition with subsequent heat treatment at moderate temperatures. Victor and Alain developed a generic theoretical framework for calculating surface coverage by a solid film of material de-wetting on the substrate. Using experimental data from OMHP thin films as an example, they calculated surface coverage for a wide range of annealing (heating) temperatures and film thicknesses. Their model accurately reproduced solution-cast thin film coverage (see the figure) and identified methods for both high and low levels of surface coverage.

Expanding their research Victor and Alain have also looked at the kinetics of photo-generated charge carriers in OMHP. By modelling the time decay of photoluminescence in the material, they extracted important information about charge carrier lifetime and concentration of intrinsic point defects. The latter are highly detrimental for the photo-voltaic performance of OMHP. The models, together with the experimental studies of colleagues, clarified the origin of the point defects and consequently identified a means of significantly decreasing their concentration.

Victor and Alain’s research can be explored in more detail on Victor and Alain's homepages. Their work reiterates the central importance of mathematical modelling in addressing real world problems.

Thursday, 19 January 2017

The impact of mathematics – human interactions!

Think of a mathematician and you might imagine an isolated individual fueled by coffee whose immaculate if incomprehensible papers may, in the fullness of time, via a decades-long dry chain of citations, be made use of by an industrialist (via one or two other dedicated mathematicians).

Not so, says new research by Oxford Mathematician and Computer Scientist Ursula Martin and evaluator Laura Meagher. Instead they reveal a vibrant and fertile environment where human interaction is the key.

Mathematics’ impact in every walk of life is astounding. Deloitte estimate that 10% of all UK jobs and 16% of total UK GDP is a direct result of mathematics. Ursula and Laura’s research puts the flesh on those figures, literally so as it demonstrates that mathematical impact is brought about above all by human interaction, long term relationships and close working with other disciplines and end users.  

In the context of an increasing interest in generating and measuring impacts across the academic and funding worlds, Ursula and Laura used the trove of data provided by the 2014 Research Excellence Framework (which assesses the quality and impact of research across higher education in the UK) to dig down into 209 examples of the impact of UK mathematics and statistics.

Complementing this with surveys and in-depth interviews, they identified a diverse ecosystem of people and ideas across mathematics, an ecosystem that includes the many other disciplines where mathematics is crucial as well as the many end users and beneficiaries of mathematical research. Their research also highlighted the role of specialist individuals in building long term relationships.

Moreover, the varieties of impact were striking, both the deep conceptual work that can reshape a whole field, and the detailed deployment of that work in a specific problem domain, both mathematical and beyond.

Finally, their work reinforced the crucial role of universities in developing a culture supportive of impact generation which reinforces the distinctive but all-pervasive nature of  mathematics, a discipline that is underpinning and influencing so many of the scientific, technological and social questions we are asking of our world.


Wednesday, 18 January 2017

Why the Truth Matters. Tim Harford's Oxford Mathematics Public Lecture 8 February

In our latest Public Lecture Tim Harford, Financial Times columnist and presenter of Radio 4's "More or Less", argues that politicians, businesses and even charities have been poisoning the value of statistics and data. Tim will argue that we need to defend the value of good data in public discourse, and will suggest how to lead the defence of statistical truth-telling.

8 February, 4pm, Mathematical Institute, Oxford. Please email to register




Monday, 16 January 2017

The magic of numbers - finding structure in randomness

Mathematics is full of challenges that remain unanswered. The field of Number Theory is home to some of the most intense and fascinating work. Two Oxford mathematicians, Ben Green and Tom Sanders, have recently made an important breakthrough in an especially tantalising problem relating to arithmetic structure within the whole numbers.

Imagine colouring every positive whole number with one of three colours, say red, green and blue. You might end up with this colouring: R1, G2, B3, B4, G5, R6, B7, R8, R9, G10, B11, B12, R13, G14, B15, B16, R17 ...

Or perhaps you chose one of the other infinitely many possibilities. Can you always pick two of these numbers, say x and y, so that x, y, x+y and xy all have the same colour?

For example, in our choice of colouring above we see that 3, 4, 3+4=7 and 3x4=12 are all blue, so it's possible for this colouring.  But will we always be able to do this, regardless of the colouring? And what if we use four colours, or even more, will we always be able to do it then too?

This is a well-known question in Ramsey theory, a branch of combinatorics that seeks to establish the existence of structure (such as the pattern x, y, x+y and xy all having the same colour) in randomness (such as all the many colourings). Results in Ramsey theory link with other areas of mathematics, and also have applications in other fields, notably in theoretical computer science.

A hundred years ago, the mathematician Issai Schur showed that if we colour every positive whole number with one of three or more colours then there are always two numbers, say x and y, so that x, y and x+y all have the same colour.  Extending this to handle multiplication as well as addition has turned out to be a significant challenge.

Ben and Tom have solved a sort of 'model' problem, where instead of colouring the integers they work with a finite analogue. This approach of considering a model problem has proved extremely fruitful for a number of other related questions, but it was far from clear how to proceed in this case. Ben and Tom introduced a number of new techniques in order to solve the problem in this finite model situation.

How does the finite analogue work?  Instead of colouring the positive whole numbers, Green and Sanders use a different system that shares similar arithmetic properties. Imagine a clock that shows 7 hours (rather than the familiar 12).  Every 7 hours, the hour hand gets back round to the top.  We can do arithmetic in this scenario too, for example 13 + 17 = 6 + 3 = 2 in this world, and 13 x 17 = 6 x 3 = 4. Ben and Tom showed that if the 7 values in this system are coloured using three or more colours then there must be many x and y for which x, y, x+y and xy all have the same colour, and similarly for any such system where the number of values is prime. Their work has been published in the new journal Discrete Analysis.

Indeed there has already been follow-up work, with Ben and his Oxford Graduate Student Sofia Lindqvist using similar ideas to resolve a question about monochromatic solutions to the equation $x+y=z^2$.

The original problem may remain unanswered, but mathematicians now have a way in and are working to establish whether Ben and Tom’s work provides the answer, both to this and maybe further problems in number theory. As so often, successes are incremental and hard-fought as mathematicians continue to map the structures that populate their and our world.