Monday, 8 January 2018

Oxford Mathematics Research probes the inner secrets of L-functions

Oxford Mathematician Tom Oliver talks about his research in to the rich mine of mathematical information that are L-functions.

"I am interested in the analytic properties of L-functions, which are supposed to encode deep arithmetic information. The best known L-function is the Riemann zeta function, which is intimately connected to the theory of prime numbers. More general L-functions are attached to algebraic varieties and automorphic forms - the former being solutions to systems of polynomials, and the latter being analytic functions on certain symmetric spaces of arithmetic interest. An elaborate web of conjectures, known as the Langlands programme, relates the two.

The L-functions of automorphic forms have well-documented analytic behaviour. My recent research attempts to answer the converse question - if we know an L-function has (weakened versions of) such analytic properties, is there an underlying automorphic form? The aim of the game is to make the assumptions as weak as possible. In doing so, we open the door to various applications. For example, the so-called 'grand simplicity hypothesis' suggests that the non-real zeros of automorphic L-functions are simple and linearly independent (over the field of rational numbers). It is exceptionally difficult to prove anything like this in wide generality, so I am interested in special cases.   

In order to get a handle on what is achievable, one has to understand ways in which L-functions can be more or less complicated than one another. A common way to quantify the complexity of an L-function is a numerical invariant called the degree. Intuitively, this is the number of gamma functions appearing in the functional equation. For example the Riemann zeta function has degree 1, and the degree grows with the degree of the base field, the dimension of the algebraic variety, the rank of the algebraic group, etc. The higher the degree, the more 'trivial' zeros an L-function has. These are the zeros on the real line cancelling the poles of the gamma function.

In a recent collobaration with Michalis Neururer (TU Darmstadt), we studied the simplest proxy for independence of zeros imaginable, which is still unresolved even for small degrees. Namely, when dividing two L-functions, one would expect that the quotient has infinitely many poles (zeros of the denominator which are not zeros of the numerator). This is easily seen to be the case when the degree of the numerator is less than that of the denominator, as the denominator has more trivial zeros. These are the negative degree quotients. More interesting are quotients of positive degree, where the poles correspond to non-trivial zeros. 

In 2013, Andrew Booker proved that degree 0 and degree 1 quotients have infinitely many poles. Building upon Booker's ideas, our work focused on the case that the difference in degree is 2. Assuming that the quotient has only finitely many poles, we show that the quotient looks enough like an automorphic L-function (in fact, the L-function of a Maass form), to apply a suitably general converse theorem. This gives rise to a factorisation of the numerator, contradicting orthogonality results for automorphic L-functions. The argument involves the construction of special symmetries of the hyperbolic plane."  

Thursday, 4 January 2018

Scaling the Maths of Life - Oxford Mathematics Public Lecture, 7 February

Oxford Mathematics Public Lectures

Scaling the Maths of Life - Michael Bonsall

In this talk Michael Bonsall will explore how we can use mathematics to link between scales of organisation in biology. He will delve in to developmental biology, ecology and neurosciences, all illustrated and explored with real life examples, simple games and, of course, some neat maths.

Michael Bonsall is Professor of Mathematical Biology in Oxford.

7 February 2018, 5pm-6pm, Mathematical Institute, Oxford.

Please email to register.

Wednesday, 3 January 2018

Why your morning cup of coffee sloshes

Americans drink an average of 3.1 cups of coffee per day (and mathematicans probably even more). When carrying a liquid, common sense says walk slowly and refrain from overfilling the container. But easier said than followed. Cue sloshing.

Sloshing occurs when a vessel of liquid—coffee in a mug, water in a bucket, liquid natural gas in a tanker, etc. - oscillates horizontally around a fixed position near a resonant frequency; this motion occurs when the containers are carried or moved. While nearly all transport containers have rigid handles, a bucket with a pivoted handle allows rotation around a central axis and greatly reduces the chances of spilling. Although this is not necessarily a realistic on-the-go solution for most beverages, the mitigation or elimination of sloshing is certainly desirable. In a recent article published in SIAM Review, Oxford Mathematicians Hilary and John Ockendon use surprisingly simple mathematics to develop a model for sloshing. Their model comprises a mug on a smooth horizontal table that oscillates in a single direction via a spring connection. “We chose the mathematically simplest model with which to understand the basic mechanics of pendulum action on sloshing problems,” John said.

The authors derive their inspiration from a Nobel prize-winning paper describing a basic mechanical model that investigates the results of walking backwards while carrying a cup of coffee. They use both Newton’s laws of physics and the basic properties of hydrodynamics to employ a so-called “paradigm” configuration, which explains how a cradle introduces an extra degree of freedom that in turn modifies the liquid’s response. “The paradigm model contains the same mechanics as the pendulum but is simpler to write down,” Ockendon said. “We found some experimental results on the paradigm model, which meant we could make some direct comparisons.”

The authors evaluate this scenario rather than the more realistic but complicated use of a mug as a cradle that moves like a simple pendulum. To further simplify their model, they assume that the mug in question is rectangular and engaged in two-dimensional motion, i.e., motion perpendicular to the direction of the spring’s action is absent. Because the coffee is initially at rest, the flow is always irrotational. “Our model considers sloshing in a tank suspended from a pivot that oscillates horizontally at a frequency close to the lowest sloshing frequency of the liquid in the tank. Together we have written several papers on classical sloshing over the last 40 years, but only recently were we stimulated by these observations to consider the pendulum effect.”

Variables in the initial model represent (i) a hand moving around a fixed position, (ii) the frequency of walking, typically between 1-2 Hertz, and (iii) a spring connecting the shaking hand to the mug, which slides on the table’s smooth surface. Hilary and John are most interested in the spring’s effect on the motion of the liquid. 

The authors solve the model’s equations via separation of variables and analyse the subsequent result with a response diagram depicting the sloshing amplitude’s dependence on forcing frequency. The mug’s boundary conditions assume that the normal velocity of both the liquid and the mug are the same, and that the oscillation’s amplitude is small. Hilary and John linearize the boundary conditions to avoid solving a nonlinear free boundary problem with no explicit solution. They record the equation of motion for the container to couple the motion of the liquid and the spring. In this case, the spring’s tension and the pressure on the walls of the container are the acting horizontal forces.

The authors discover that including a string or a pendulum between the container and the carrying hand (the forcing mechanism) lessens the rigidity and dramatically decreases the lowest resonant frequency, thus diminishing sloshing for almost all frequencies. “Our model shows that, compared to an unpivoted tank, the amplitude of the lowest resonant response will be significantly reduced, provided the length of the pendulum is greater than the length of the tank.” 

In conclusion, Hilary and John use simplistic modeling and analysis to explain a common phenomenon that nearly everybody experiences. They suggest that future analysts investigate sloshing in a cylindrical rather than rectangular mug, or with vertical rather than horizontal oscillations, as both of these factors complicate the model. One could also examine the spring action’s effect on the system’s nonlinear behavior near resonance. Ultimately, researchers can employ basic ideas from this study to consider the nonlinear response of shallow water sloshing, which has a variety of real-world applications.

A version of this article by Linda Sorg first appeared in SIAM news.

Sunday, 31 December 2017

What do fireflies and viruses have in common?

Oxford Mathematician Soumya Banerjee talks about his current work in progress.

"On warm summer days, fireflies mesmerise us with their glowing lights. They produce this cold light using a light-emitting molecule, the luciferin, and a complementary enzyme, luciferase. This process is known as bioluminescence.

Scientists have now genetically engineered this process in viruses. After infecting cells, these modified viruses (called replicons) produce light using the firefly genes. Recently this was used to study how West Nile virus (similar to Zika virus) infects cells.

I have now developed a mathematical model to analyse how firefly genes produce luminescence in these virus infected cells. The model predicts how the luminescence or brightness would gradually decrease as the cells infected by the virus are slowly killed off. This was then matched to experimental data.

The mathematical models predict that some cells in the lymph node live for about 12 hours after being infected with the virus. These cells also release the most virus into the blood. The work suggests that these particular cell types can be targeted using therapies such as anti-viral medication to fight the infection."

You can read more about the work which is at pre-publication stage here.

Tuesday, 19 December 2017

How understanding Oscillator Networks could help unlock the secrets of brain diseases

Oxford Mathematician Christian Bick talks about his and colleagues' research into oscillator networks and how it could be valuable in understanding diseases such as Parkinson's.

"Many systems that govern crucial aspects of our lives can be seen as networks of interacting oscillators. On a small scale, for example, the human brain consists of individual cells that can send bits of information to each other periodically. On a large scale, the power grid of an entire country can be seen as a network of rotating units (generators and motors). The function of these oscillator networks crucially depends on how the units evolve together. For a power grid to be stable it needs to be synchronized to a common grid frequency—think of the 50Hz coming out of  your power outlets at home. By contrast, too much synchronization in the brain is believed to be detrimental as it has been associated with a range of disorders such as epilepsy and Parkinson’s disease.

Synchronization, where distinct units will behave in the same way as time goes on, can arise spontaneously in networks of interacting oscillators. Already Christian Huygens noticed in 1665 that two of his oscillating pendulum clocks showed an “odd kind of sympathy” when they were allowed to interact: they would swing in unison after some time. Mathematical models can help understand how network interactions—the coupling between oscillators—allow for synchronization to arise in oscillator networks.

In contrast to synchronized dynamics, we are interested in networks where identical oscillators can show distinct dynamics in the following sense. Take a collection of oscillators which would oscillate at the same frequency in isolation. Now when you make these oscillators interact in a particular way, then half of the oscillators will be oscillating at one frequency while the other half is oscillating at another frequency! In other words, there is a separation of frequencies that purely arise due to the network interactions. In work with collaborators at St. Louis University we studied this phenomenon in a network of four oscillators. We first analyzed the effect in a mathematical model. Guided by the theoretical work, we then demonstrated the same effect in an experimental setup of electrochemical oscillators with suitable network interactions.

Our results shed further light on the effect the network structure has on the dynamics of interacting oscillators. In particular, they indicate what network interactions allow for dynamics where only part of the oscillators are synchronized in frequency. Hence, if neural disease is indeed related to abnormal levels of synchrony, these insights could, for example, be useful in devising means to prevent or counteract pathological synchronization (as in Parkinson's disease) by tuning the network interactions."

Thursday, 14 December 2017

Alex Bellos Oxford Mathematics Christmas Public Lecture now online

In our Oxford Mathematics Christmas Public Lecture Alex Bellos challenges you with some festive brainteasers as he tells the story of mathematical puzzles from the Middle Ages to modern day.

Alex is the Guardian’s puzzle blogger as well as the author of several works of popular maths, including Puzzle Ninja, Can You Solve My Problems? and Alex’s Adventures in Numberland.








Monday, 11 December 2017

Philip Maini honoured by Indian National Science Academy

Oxford Mathematician Professor Philip Maini FRS, Professorial Fellow in Mathematical Biology at St John’s College has been elected a Foreign Fellow of the Indian National Science Academy for his mathematical and computational modelling of biological processes relevant to wound healing and vascular tumour growth, scar formation and cancer therapy. Philip's previous work has included influencing HIV/AIDS policy in India through mathematical modelling. The election will be effective from 1 January 2018.

Wednesday, 6 December 2017

Oxford Mathematics Virtual Open Day for Masters' Courses, TODAY, Thursday 7 December, 3pm

Today, Thursday 7th December 2017, Oxford Mathematics will be holding its second Graduate Virtual Open Day, from 15:00-16:00 (UK time). This year, the Virtual Open Day will be focusing on taught masters' courses offered at the Mathematical Institute, which will include the following degrees:

MSc Mathematical and Computational Finance
MSc Mathematics and Foundations of Computer Science
MSc Mathematical Modelling and Scientific Computing
MSc Mathematical Sciences
MSc Mathematical and Theoretical Physics

This will be an interactive livestreamed event, where members of faculty will be providing information on the courses mentioned above and also will be answering your queries. If you are a prospective applicant, please e-mail your questions to and tweet them to @OxUniMaths and we will attempt to answer as many of these questions during the hour as possible.


Monday, 4 December 2017

Andrew Wiles London Public Lecture now online

In the first Oxford Mathematics London Public Lecture, in partnership with the Science Museum, world-renowned mathematician Andrew Wiles lectured on his current work around Elliptic Curves followed by an-depth conversation with mathematician and broadcaster Hannah Fry.

In a fascinating interview Andrew talked about his own motivations, his belief in the importance of struggle and resilience and his recipe for the better teaching of his subject, a subject he clearly loves deeply.









Friday, 1 December 2017

Modelling outbreaks of infectious disease - diagnostic tests key for epidemic forecasting

Precise forecasting in the first few days of an infectious disease outbreak is challenging. However, Oxford Mathematical Biologist Robin Thompson and colleagues at Cambridge University have used mathematical modelling to show that for accurate epidemic prediction, it is necessary to develop and deploy diagnostic tests that can distinguish between hosts that are healthy and those that are infected but not yet showing symptoms. The data derived from these tests must then be integrated into epidemic models.

“We used Ebola virus disease as the main case study in this paper" says co-author Nik Cunniffe, "since at the date of publication it was an important and very timely example of the type of disease we focus on in our research, i.e. one for which reporting is incomplete and where some epidemics die out naturally before infecting a large number of people.”

Robin is currently working on a probabilistic modelling framework for managing outbreaks of diseases such as bovine tuberculosis and foot-and-mouth disease: “I am investigating the optimal time to introduce control of a newly invading pathogen. Early control can be beneficial since the outbreak might be suppressed before the pathogen sweeps through the population. However, later control carries the advantage that it allows transmission parameters to be estimated more accurately and interventions to be optimised. Deciding when to initiate control is therefore an optimal stopping problem, and involves balancing the benefits of waiting against the potential costs of the pathogen becoming widespread.”

The team have won the PLoS Computational Biology Research Prize 2017 for the public impact of their work about diagnostic testing for Ebola. Robin is now a Junior Research Fellow at Christ Church in Oxford. He undertook this project as part of his PhD studies in Cambridge.