Monday, 9 September 2019

Elementary particles in Flatland

Elementary particles in two dimensional systems are not constrained by the fermion-boson alternative. They are so-called "anyons''. Anyon systems are modelled by modular tensor categories, and form an active area of research. Oxford Mathematician André Henriques explains his interest in the question.

"In the standard model of quantum physics, which describes our world at subatomic scales, the elementary particles are divided into two kinds. The fermions on the one hand form the matter that is present all around us (examples include electrons, quarks, neutrinos, etc.). And the bosons are responsible for the transmission of the four fundamental forces (e.g. photons, gluons, the Higgs boson, etc.).

Mathematically, one can devise alternative universes with different laws of physics, in which the number of elementary particles is different from that of our physical world. But the fundamental dichotomy between bosons and fermions is dictated by the $3$-dimensionality of the space we live in. Fermions are defined by the property that, under exchanging two particles of the same kind, the wave-function of the many body system acquires a minus sign, while for bosons, that same operation leave the wave-function unchanged.




The number which appears above is constrained to be $\pm1$ because, in $3$ dimensions, the operation of twice exchanging two particles is homotopic to the operation of doing nothing. That is, there exists a one-parameter family of operations where, at one end of the family you have the operation of twice exchanging the particles, and at the other end of the family you have the operation of doing nothing:




The existence of that homotopy forces the above number to square to 1, which leaves us only +1 or -1 as possible values.

Let us now switch to flatland, in which space is 2-dimensional. The operations drawn in the above pictures are no longer homotopic, and the elementary excitations (quasi-particles) are no longer constrained to the Fermi-Bose alternative. They can be anything interpolating between fermions and bosons, and have thus been called "anyons''. The typical approach to mathematically formalise the notion of anyon systems is that of modular tensor categories. The objects of the modular tensor category represent the different species of anyons, and the tensor product of the object in the modular tensor category corresponds to the operation of letting two (or more) anyons collide into each other.

The construction of modular tensor categories is a generally difficult task, and every time one finds a new source of examples, this is treated as a nice result. An important class of examples come from the so-called quantum groups. These are certain Hopf algebras closely related to the classical Lie groups (e.g. the group of orthogonal rotations of $n$-dimensional space). The modular tensor categories coming from quantum groups have a very nice structure: their set of simple objects (= anyon types) are in bijection with the lattice points inside a simplex.




The simplest but already quite interesting example of a modular tensor category is known as the Fibonacci modular tensor category. It has a single non-trivial anyon type. The latter has the property that, by letting n identical anyons collide, the number of possible quantum states of the output is exactly equal to the n-th Fibonacci number. Finding new examples of modular tensor categories is an active field of research, and there exists an important conjecture which states that every modular tensor category is related, in a specific way, to quantum groups.

In my study of the representation theory of certain infinite dimensional groups, I have found a new construction of the quantum group modular tensor categories. They are constructed as centres of categories of representations of the based loop group. In the future, I hope to apply based loop group techniques to other structures in 2-dimensional physics, such as boundary conditions, and line defects."

Wednesday, 28 August 2019

Nick Trefethen elected to the Academia Europaea

Oxford Mathematican Nick Trefethen has been elected to the Academia Europaea. Nick is Professor of Numerical Analysis in Oxford, a Fellow of Balliol College and Head of Oxford Mathematics's Numerical Analysis Group. He has published around 140 journal papers spanning a wide range of areas within numerical analysis and applied mathematics, including non-normal eigenvalue problems and applications, spectral methods for differential equations, numerical linear algebra, fluid mechanics, computational complex analysis, and approximation theory.

Nick will be an invited speaker at the 8th European Congress of Mathematics in Slovenia in 2020.

Friday, 23 August 2019

Minimal Lagrangians and where to find them

Minimal Lagrangians are key objects in geometry, with many connections ranging from classical problems through to modern theoretical physics, but where and how do we find them?  Oxford Mathematician Jason Lotay describes some of his research on these questions.

"A classical problem in geometry going back at least to Ancient Greece is the so-called isoperimetric problem: what is the shortest curve in the plane enclosing a given area A?  The answer is a circle:



Variants of this problem for curves include finding the shortest curve representing a given class of loops. For example on a torus, we see that both the blue and red curves on the right are shortest curves which represent the original blue loop on the left.



If we look at a sphere, however, then there is no shortest curve representing any given loop, since we can contract every curve to a point.  A natural question on a sphere is to ask for the shortest curve dividing the sphere into regions of equal area, which sounds more like the original isoperimetric problem.  The answer is (any) equator:



The answer for the shortest curve in every case we have considered has in fact been a minimal Lagrangian: on surfaces, minimal Lagrangians are just geodesics, which are curves that are critical for length.  Moreover, deformations of curves in the plane which contain the same enclosed area, or of the curves on the sphere which always divide the sphere into regions of equal area, are known as Hamiltonian isotopies.  In particular, we can rephrase the problem we considered on the sphere as finding minimal Lagrangians in the Hamiltonian isotopy class of the original blue curve on the left sphere.

These ideas generalize to higher dimensions: we replace the surface by a particular type of even-dimensional space known as a Kähler-Einstein manifold M, we replace curves by Lagrangians that are certain geometric objects in M which are half the dimension of M (just like curves are 1-dimensional whilst surfaces are 2-dimensional), and Hamiltonian isotopies are special types of deformations we allow of a given Lagrangian that generalize the area preserving deformations we had before.  Then minimal Lagrangians are just Lagrangians which are critical for volume, just as geodesics were curves which are critical for length, so we can think of them a bit like soap films:



Kähler-Einstein manifolds and minimal Lagrangians form an important part of modern geometry, particularly since they include so-called Calabi-Yau manifolds and special Lagrangians, which play a key role in the study of Mirror Symmetry and in String Theory in theoretical physics.  The definition of Kähler-Einstein manifolds depends on a constant, which can be positive, zero (which is the Calabi-Yau setting) or negative, and the behaviour of these spaces and the minimal Lagrangians in them is quite different in each case.

The positive Kähler-Einstein manifolds include complex projective spaces, which generalize the sphere, and here minimal Lagrangians are typically easy to find because the ambient space has a lot of symmetries (like the sphere).  An interesting question therefore is: which minimal Lagrangian in a positive Kähler-Einstein manifold has the least volume in its Hamiltonian isotopy class?  For the sphere and curves which divide the sphere into two regions of equal area, the answer is the equator.  There is a well-known and unsolved conjecture, due to Oh, that says for complex projective spaces there is a natural minimal Lagrangian (called the Clifford torus) which is volume-minimizing under Hamiltonian isotopies: this is the higher-dimensional analogue of our isoperimetric problem on the sphere.

A natural way to try to solve Oh’s conjecture is to start with a Lagrangian which is related to the Clifford torus by a Hamiltonian isotopy and then deform the Lagrangian by minimizing volume as quickly as possible: this is called Lagrangian mean curvature flow.  I have studied Lagrangian mean curvature flow extensively, most recently with my collaborators Ben Lambert (Oxford) and Felix Schulze (UCL), with a hope to try to tackle this and other related problems.

In particular, in the Calabi-Yau setting (which is, in some sense, the analogue of the flat plane or the torus), Lagrangian mean curvature flow is particularly appealing and there is a conjecture due to Thomas and Yau that predicts the behaviour of the flow in special cases.  I am actively studying approaches to this conjecture, which would hopefully pave the way towards understanding how we can potentially “break up” a Lagrangian into special Lagrangian pieces, in the same way that we now know how to break up 3-dimensional spaces into special (or “prime”) pieces using the geometric flow known as the Ricci flow.

Finally, in negative Kähler-Einstein manifolds (which are the higher-dimensional analogues of surfaces with more 'holes' than the torus), minimal Lagrangians are typically very hard to find.  In fact, they are conjectured to be unique in their Hamiltonian isotopy class.  This is obviously false for positive Kähler-Einstein manifolds since, for example, on the sphere we can rotate the sphere to get the red geodesic starting from the blue one, which is a Hamiltonian isotopy.



That said, there are many examples of negative Kähler-Einstein manifolds, so it is natural to ask whether they contain minimal Lagrangians or not.  Together with my collaborator Tommaso Pacini (Turin), I recently addressed this question by finding infinitely many new cases of negative Kähler-Einstein manifolds containing minimal Lagrangians.

Although we have made a lot of progress in understanding minimal Lagrangians, there are many fascinating and challenging problems left to study.  Minimal Lagrangians certainly are fantastic beasts and we should try our best to find them!"

For more on Jason's work:
Ancient solutions in Lagrangian mean curvature flow
Remarks on the self-shrinking Clifford torus
From minimal Lagrangian to J-minimal submanifolds: persistence and uniqueness

Thursday, 15 August 2019

PROMYS Europe celebrates its 5th birthday

2019 sees the 5th PROMYS Europe summer school.  The programme brings together enthusiastic and ambitious teenage mathematicians from across Europe, who gather in the Oxford Mathematical Institute for six weeks of intensive mathematics.  Participants, who stay at Wadham College, work on activities designed to give them the opportunity to explore mathematical ideas independently.  This year they are concentrating on number theory and combinatorics, and in addition are working on group projects drawing on ideas from a range of mathematical topics.  The programme is a partnership of the Oxford Mathematical Institute, Wadham College, the Clay Mathematics Institute, and PROMYS in Boston, which celebrated its 30th birthday this year.

PROMYS Europe, like its parent programme PROMYS in the USA, has a distinctive teaching philosophy and structure.  Students receive a daily set of problems, and have a daily number theory lecture, but the lectures aim to be at least three days behind the problems sets.  Students are invited to experiment, to gather numerical data, to explore ideas, to formulate conjectures and to try to find their own proofs, all before the ideas are formalised in lectures.  This gives students a very different experience of mathematics from anything they have encountered previously, and they are able to see how deeply it is possible to understand an area of maths because they have put the ideas together themselves. 

Students apply for the programme by submitting their work on our challenging application problems.  Students need to display perseverance and creativity in testing their ideas and finding routes to a solution.  To join the email list to be notified when applications for students and counsellors open next year, please see the PROMYS Europe website.

Halfway through the programme, the 2019 students and undergraduate counsellors gathered with faculty, former students and counsellors, and friends of the programme, to celebrate the 5th birthday of PROMYS Europe and the many achievements of the students and counsellors.  Many former students and counsellors of PROMYS Europe have gone on to study maths degrees at leading universities, and the oldest are in some cases now embarking on PhD research degrees.

This year, the 36 students and counsellors come from Belgium, Bulgaria, Czech Republic, Finland, Germany, Hungary, Poland, Romania, Serbia, Slovakia, Spain, Sweden, Switzerland, Turkey, UK and Ukraine.  The counsellors are all Mathematics undergraduates, including three who are current Oxford undergraduates (having previously themselves been students at PROMYS Europe), and a fourth who is moving to Oxford in October to start the Oxford Masters in Mathematical Sciences (OMMS) MSc.

PROMYS Europe is dedicated to the principle that no one should be unable to attend for financial reasons, and full or partial financial assistance is available for those who would otherwise be unable to attend.  PROMYS Europe is made possible thanks to funding and other resources provided by the partnership, as well as further financial support from alumni of the University of Oxford and Wadham College, and from the Heilbronn Institute for Mathematical Research.

Thursday, 18 July 2019

Using maths for infectious disease epidemic forecasting

The 1918 Spanish influenza pandemic claimed around fifty million lives worldwide. Interventions were introduced to reduce the spread of the virus, but these were not based on quantitative assessments of the likely effects of different control strategies. One hundred years later, mathematical modelling is routinely used for forecasting and to help plan interventions during outbreaks in populations of humans, animals and plants.

In two recent linked theme issues of the journal Philosophical Transactions of the Royal Society B, Oxford Mathematician Dr Robin Thompson and Dr Ellen Brooks-Pollock (University of Bristol) commissioned articles about modelling epidemics in humans, animals and plants. While there are differences between pathogens in these different types of host population, there are also a number of similarities between the questions that modelling is used to address.

As an example, when a pathogen first arrives in a host population, mathematical models can be used to assess whether or not initial cases will fade out, or whether they will lead on to a major epidemic. This analysis can in fact be performed using only a quadratic equation!

Probability of a major epidemic

To estimate the risk of a major epidemic when a pathogen first arrives in a population, we denote by $q_I$ the probability that a major epidemic does not occur, starting from $I$ infected individuals. The probability of a major epidemic when the pathogen first arrives in the population (i.e. starting from one infected individual) is therefore $1 - q_1$.

Conditioning on whether the first individual infects another host, or instead recovers, leads to the equation \begin{eqnarray*} q_1 = \text{P} {\rm (infection)} q_2 + \text{P} {\rm (recovery)} q_0. \end{eqnarray*} The variable $q_0$ represents the probability that a major epidemic does not occur starting from $0$ infected hosts. If there are no infected hosts, then a major epidemic will certainly not occur, so $q_0 = 1$. A major epidemic failing to occur from two infected hosts requires no major epidemic from either infected host (and for each of these hosts, the probability of no major epidemic is $q_1$). If infection lineages from the two infected individuals are independent of each other, then $q_2 = {q_1}^2$.

As a result, the probability of a major epidemic starting from one infected host is $1 - q_1$, where $q_1$ is the minimal solution of the quadratic equation \begin{eqnarray*} q_1= \text{P} {\rm (infection)} {q_1}^2 + \text{P} {\rm (recovery)}. \end{eqnarray*} Calculations like these are increasingly used for epidemic forecasting. For example, the probability of a flare up of Ebola in Nigeria was estimated in this way during the 2014-16 Ebola epidemic in West Africa. 

Mathematical models will continue to play important roles for forecasting and guiding interventions during epidemics. To find out more, check out this review article or contact Robin.

(The image above is a visualisation of air traffic routes over Eurasia, which could be used to inform models of global pathogen transmission. Credit: Globaïa, 2011).

Thursday, 11 July 2019

Oxford Mathematicians win teaching and equality and diversity awards

Oxford Mathematics is part of the Mathematical, Physical and Life Sciences Division (MPLS) here in Oxford and every year the division gives teaching and equality and diversity awards in recognition of the fact that teaching and the learning environment are at the very core of what we are about and from where all future success will derive.

This year Oxford Mathematicians were successful in various ways. In the teaching category Mareli Grady (who splits her time with the Department of Statistics) won for raising awareness of mathematics and engagement with the public through the Oxford Maths Festival; while Ian Griffiths, Sam Cohen and Frances Kirwan were recognised for their work on Fridays@4, an initiative which since 2015-16 has enhanced graduate students' study skills and their long-term educational development, and helped integrate students within the department.

Under the equality and diversity heading the Outstanding Contribution by a Staff Member – Student Choice Award is made to an individual (academic, researcher or administrator), nominated by a student or group of students, who has made an important contribution to advancing equality and diversity. Dominic Vella from Oxford Mathematics was nominated for creating and fostering a diverse and inclusive research group, with people from many different countries, socio-economic and educational backgrounds, and varying ethnic, gender and age profile.


Monday, 8 July 2019

Siddharth Arora and colleagues win Martin Black Prize for 2019 from the Institute of Physics and Engineering in Medicine

The Institute of Physics and Engineering in Medicine has announced the winners of the Martin Black Prize for 2019, which is awarded for the best paper published in Physiological Measurement.

The winner was 'Big data in Parkinson’s disease: using smartphones to remotely detect longitudinal disease phenotypes’ by Oxford Mathematician Siddharth Arora and John Prince and Maarten de Vos from Oxford's Institute of Biomedical Engineering and Department of Engineering Science respectively.

The paper's objective was to better understand the longitudinal characteristics of Parkinson’s disease through the analysis of finger tapping and memory tests collected remotely using smartphones. More on Siddharth's work can be found here.

Tuesday, 2 July 2019

Oxford Mathematics trio scoop LMS Prizes

Three Oxford Mathematicians have been awarded 2019 London Mathematical Society (LMS) Prizes.

Andrew Wiles has been awarded a De Morgan Medal for his seminal contributions to number theory and for his resolution of ‘Fermat’s Last Theorem’ in particular, as well as for his numerous activities promoting mathematics in general.

Ben Green has been awarded a Senior Whitehead Prize for his groundbreaking results in additive combinatorics, analytic number theory and group theory.

David Conlon has been awarded a Whitehead Prize in recognition of his many contributions to combinatorics. His particular expertise is Ramsey Theory, where he has made fundamental contributions to both the arithmetic and graph-theoretic sides of the subject. 

Tuesday, 2 July 2019

Oxford Mathematics Open Day LIVE JULY 3, 10am

Whether you are interested in Maths at Oxford. Whether you are interested in Maths. Whether you are just interested. The Oxford Mathematics Open Day is LIVE on July 3 @10am. You can watch it all, or just dip in and out. You can also ask questions in our live chat at 10.30am. Virtually come along. You are very welcome.


The films will remain available after the live stream and will join the student lectures and tutorial which are proving very popular on our Oxford Mathematics YouTube Channel.

Friday, 21 June 2019

Sticking with droplets: How having a soft foot can improve capillary adhesion

Have you ever picked up a glass to find that the coaster it was resting on remains stuck to the bottom? If so, then you have experienced the ability of fluid to stick two surfaces together. When the bottom of the glass is wetted, for example by accidentally spilling a drink, then this fluid can fill the gap between the glass and coaster. The surface tension of the liquid then provides a pulling force on the coaster that keeps it attached to the glass.

It is believed that the climbing ability of insects is aided by this effect, albeit at a smaller scale. Insects are able to walk up walls and across ceilings, and have been observed holding over 100 times their own body weight while remaining adhered. Looking closely beneath their feet, an oily fluid is found that is believed to help stick their feet to a substrate. It has also been noted that the footpads of some insects are soft and deformable.

Inspired by these observations of insects, Oxford Mathematicians Matthew Butler and Dominic Vella have developed a mathematical model to investigate the combination of deformability and adhesion. They consider a system where the footpad is a thin, deformable sheet under tension (like a drum skin) in which the deformability can be varied by changing the tension in the sheet, and the separation of the solid surfaces can be controlled.

Calculations of the equilibria reveal two distinct stable states, distinguished by whether the deformable sheet is in contact with the substrate or not. In both stable states, the adhesion force is found to be always larger than an equivalent rigid case, suggesting that deformation can be good—the surface tension deforms the sheet, increasing the force provided by surface tension: a positive feedback. Further, when the sheet is in contact with the substrate, the adhering force can be orders of magnitude larger than the non-contacting state.

Experiments performed by Finn Box and Thomas Robert show a good qualitative agreement with the theory, as well as demonstrating that adhesion is possible in a lab scale model. However, it was found that adhesion was only maintained if the initial attachment phase took a sufficient length of time. Further study of the dynamical theory suggests this bottleneck is likely to be due to the formation of a fluid dimple as the deformable sheet tries to contact the base, but is resisted by fluid viscosity.

Despite this caveat, the significant increase in adhesion compared to the rigid case means that deformable capillary adhesives are a fruitful area for further investigation. Of particular interest with this system, is that being able to set the tension and separation may allow for high adhesion, whilst also giving easy control of detachment when required. It is hoped that further investigation of similar devices that exploit deformability may lead to advancements in the world of adhesion science.