Monday, 15 January 2018

Gauge theory - where particle physics and mathematics meet


Oxford Mathematician Yuuji Tanaka describes his part in the advances in our understanding of gauge theory.

"Gauge theory originated in physics, emerging as a unified theory of weak interaction such as appears in beta decay and electro-magnetism via the framework of Yang-Mills gauge theory together with the "Higgs mechanism" which wonderfully attaches mass to matters and forces. It became a mainstream of particle physics after the great discovery of the renormalisable property by Veltman and 't Hooft, and it gives precise descriptions of the experiments. Nowadays all fundamental interactions (electro-magnetism, weak force, strong force, gravity) can be described by gauge theory.

These development certainly stimulated the mathematical studies of gauge fields, particularly in the field of principal or vector bundles. In this context, the curvature of a connection corresponds to the field strength of a gauge field. In the early 80s Donaldson looked into the moduli space of solutions to a certain type of Yang-Mills gauge field (called self-dual or anti-self-dual connections), and produced surprising ways to distinguish differential structures of 4-dimensional curved spaces with the same topology type by using the moduli space or by defining invariants of smooth structures through the moduli space.

After Donaldson's work, Witten skilfully reinterpreted it in terms of a certain quantum field theory. Subsequently Atiyah and Jeffrey mathematically reformulated Witten's work by using the Mathai-Quillen formalism. These exchanges of ideas were one of the stepping stones which led to the discovery of the Seiberg-Witten equations and their invariants around 1994 through a generalisation of the electro-magnetic duality, a hidden symmetry in the theory of electro-magnetism. Seiberg and Witten presented a striking application of this on a super Yang-Mills theory in the quantum level, called strong-weak duality, which enables one to calculate things in the strong coupling region of the theory in terms of ones in the weak coupling region.

Vafa and Witten further analysed Seiberg and Witten's work in a more symmetric model, and conjectured that the partition function of the invariants in this case would have a modular property, which is a mathematical enhancement of the strong-weak duality mentioned above and originally discovered in the theory of elliptic curves in the 19th century. They examined this property in examples by using results from mathematics under the assumption that the Higgs fields automatically vanish.

However, even a mathematically rigorous definition of this, especially including Higgs fields, was not produced for over 20 years. Richard Thomas and myself recently defined deformation invariants of projective surfaces, from the moduli space of solutions to the gauge-theoretic equations of the Vafa and Witten theory, by using modern techniques in algebraic geometry. We then computed partition functions of the invariants coming from non-vanishing Higgs fields as well. Surprisingly, our calculations match the conjecture by Vafa and Witten more than two decades ago despite ours also including sheaves on the surface."

Click here and here for the published research.

Friday, 12 January 2018

Abel in Oxford - Lectures by Abel Prize winners and members of the Abel Prize Committee, Oxford, 15 January

The Abel Prize is the most prestigious prize in Mathematics. Each year, in anticipation of the prize announcement, an afternnon of lectues showcases previous winners and member of the Committee. This year the event will be held in Oxford on Monday 15th January. Andrew Wiles, John Rognes and Irene Fonseca will be the speakers. Full details below. Everyone welcome.  No need to register.


1.00pm: Introductory Remarks by Camilla Serck-Hanssen, the Vice President of the Norwegian Academy of Science and Letters

1.10pm - 2.10pm: Andrew Wiles

2.10pm - 2.30pm: Break

2.30pm - 3.30pm: Irene Fonseca

3.30pm - 4.00pm: Tea and Coffee

4.00pm - 5.00pm: John Rognes


Andrew Wiles: Points on elliptic curves, problems and progress

This will be a survey of the problems concerned with counting points on elliptic curves.


Irene Fonseca: Mathematical Analysis of Novel Advanced Materials

Quantum dots are man-made nanocrystals of semiconducting materials. Their formation and assembly patterns play a central role in nanotechnology, and in particular in the optoelectronic properties of semiconductors. Changing the dots' size and shape gives rise to many applications that permeate our daily lives, such as the new Samsung QLED TV monitor that uses quantum dots to turn "light into perfect color"! 

Quantum dots are obtained via the deposition of a crystalline overlayer (epitaxial film) on a crystalline substrate. When the thickness of the film reaches a critical value, the profile of the film becomes corrugated and islands (quantum dots) form. As the creation of quantum dots evolves with time, materials defects appear. Their modeling is of great interest in materials science since material properties, including rigidity and conductivity, can be strongly influenced by the presence of defects such as dislocations. 

In this talk we will use methods from the calculus of variations and partial differential equations to model and mathematically analyze the onset of quantum dots, the regularity and evolution of their shapes, and the nucleation and motion of dislocations.


John Rognes: Symmetries of Manifolds

To describe the possible rotations of a ball of ice, three real numbers suffice.  If the ice melts, infinitely many numbers are needed to describe the possible motions of the resulting ball of water.  We discuss the shape of the resulting spaces of continuous, piecewise-linear or differentiable symmetries of spheres, balls and higher-dimensional manifolds.  In the high-dimensional cases the answer turns out to involve surgery theory and algebraic K-theory.

Friday, 12 January 2018

Oxford Mathematician Sir John Ball awarded the King Faisal Prize for Science

Oxford Mathematician Sir John Ball FRS has been awarded the King Faisal Prize for Science. Launched by the King Faisal Foundation (KFF) and granted for the first time in 1979, the King Faisal Prize recognises the outstanding works of individuals and institutions in five major categories: Service to Islam, Islamic Studies, Arabic Language and Literature, Medicine, and Science. Its aim is to benefit Muslims in their present and future, inspire them to participate in all aspects of civilisation, as well as enrich human knowledge and develop mankind.

Sir John Ball is Sedleian Professor of Natural Philosophy, Director of the Oxford Centre for Nonlinear Partial Differential Equations and Fellow of the Queen's College. John's main research areas lie in the calculus of variations, nonlinear partial differential equations, infinite-dimensional dynamical systems and their applications to nonlinear mechanics.

Friday, 12 January 2018

The mathematics of security - Oxford Mathematics researchers and undergraduates expose security flaw

As recent breaches have demonstrated, security will be one of the major concerns of our digital futures. The collective intelligence of the mathematical community is critical to finding these flaws. A group of Oxford Mathematicians, both researchers and undergraduates, have done just that.

SecureRF is a corporation founded in 2004 specialising in security for Internet of Things (IoT), i.e. devices with low processing power that require ultra-low energy consumption, whose partners include the US Air Force. SecureRF has also collaborated with Intel to develop an implementation of WalnutDSA on secure field-programmable gate arrays. WalnutDSA (trademarked by SecureRF) is an example of  a digital signature algorithm, a mathematical scheme for demonstrating the authenticity of digital messages (like a real signature, but digital). Walnut DSA uses high-level mathematical techniques from permutation groups, matrix groups and braid groups, and is designed to provide post-quantum security in lightweight IoT device contexts.

The Oxford Team attacked the algorithm by bypassing the E-Multiplication and cloaked conjugacy search problems at its heart, forging signatures for arbitrary messages in approximately two minutes. Thanks to this cryptanalysis, the scheme has now been modified accordingly and an upgrade that corrects the security risk submitted to the National Insititute of Standards and Technology (NIST) competition for Post Quantum Cryptography.

One of the most pertinent and inspiring parts of the story is that the exposure was the result of a collaboration between researchers Giacomo Micheli and Christophe Petit and undergraduates Daniel Hart, DoHoon Kim, Guillermo Pascual Perez and Yuxuan Quek - the work was one of the summer projects that Oxford Mathematics uses to develop and inspire its undergraduate mathematicians, giving them a taste of rigorous research. 

A fuller explanation can be found here and will be presented at PKC 2018, the 21st edition of the International Conference on Practice and Theory of Public Key Cryptography.

Thursday, 11 January 2018

Oxford Mathematician Sarah Waters awarded a Leverhulme Trust Senior Research Fellowship

Oxford Mathematician Sarah Waters has been awarded a Royal Society Leverhulme Trust Senior Research Fellowship commencing this month. Sarah is an applied mathematician here in Oxford. Her interest is in physiological fluid mechanics, tissue biomechanics and the application of mathematics to problems in medicine and biology. Her work varies from classical applied mathematics problems motivated by physiological applications to highly interdisciplinary work - she collaborates with life scientists, clinicians, bioengineers, theoreticians and experimentalists to develop and solve models that are novel, realistic and provide insights into biomedical problems. The resulting models often lead to theoretical predictions that can be exploited in the laboratory.

Wednesday, 10 January 2018

Oxford Mathematics part of a new centre for new approaches to data science via application driven topological data analysis

Modern science and technology generate data at an unprecedented rate. A major challenge is that this data is often complex, high dimensional and may include temporal and/or spatial information. The 'shape' of the data can be important but it is difficult to extract and quantify it using standard machine learning or statistical techniques. For example, an image of blood vessels near a tumour looks very different to an image of healthy blood vessels; statistics alone cannot quantify this difference. New shape analysis methods are required.

Thanks to funding from the Engineering and Physical Sciences Research Council (EPSRC), a newly created centre combining scientists in Oxford, Swansea and Liverpool will study the shape of data through the development of new mathematics and algorithms, and build on existing data science techniques in order to obtain and interpret the shape of data. A theoretical field of mathematics that enables the study of shapes is geometry and topology. The ability to quantify the shape of complicated objects is only possible with advanced mathematics and algorithms. The field known as topological data analysis (TDA), enables one to use methods of topology and geometry to study the shape of data. In particular, a method within TDA known as persistent homology provides a summary of the shape of the data (e.g. features such as holes) at multiple scales. A key success of persistent homology is the ability to provide robust results, even if the data are noisy. There are theoretical and computational challenges in the application of these algorithms to large scale, real-world data.

The aim of this centre is to build on current persistent homology tools, extending them theoretically, computationally, and adapting them for practical applications. The Oxford team led by Heather Harrington and Ulrike Tillmann together with Helen Byrne, Peter Grindrod and Gesine Reinert is composed of experts in pure and applied mathematics, computer scientists, and statisticians whose combined expertise covers cutting edge pure mathematics, mathematical modelling, algorithm design and data analysis. This core team will in turn work closely with collaborators in a range of scientific and industrial domains.

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.