Oxford Mathematics and the Clay Mathematics Institute Public Lectures

Roger Penrose - Eschermatics

Roger Penrose’s work has ranged across many aspects of mathematics and its applications from his influential work on gravitational collapse to his work on quantum gravity. However, Roger has long had an interest in and influence on the visual arts and their connections to mathematics, most notably in his collaboration with Dutch graphic artist M.C. Escher. In this lecture he will use Escher’s work to illustrate and explain important mathematical ideas.

Professor Martin R Bridson FRS has been appointed President of the Clay Mathematics Institute from October 1, 2018. He is the Whitehead Professor of Pure Mathematics at the University of Oxford and a Fellow of Magdalen College. Until earlier this summer, he was Head of the Mathematical Institute at Oxford.

He studied mathematics as an undergraduate at Hertford College, Oxford, before moving to Cornell in 1986 for his graduate work. He completed his PhD there in 1991, under the supervision of Karen Vogtmann, with a thesis on Geodesics and Curvature in Metric Simplicial Complexes. After appointments at Princeton and at the University of Geneva, he returned to Oxford in 1993 as a Tutorial Fellow of Pembroke College. In 2002, he moved to Imperial College London as Professor of Mathematics and returned again to Oxford in 2007 as Whitehead Professor. He is a Fellow of the American Mathematical Society (2015) and a Fellow of the Royal Society (2016), to which he was elected "for his leading role in establishing geometric group theory as a major field of mathematics".

Professor Bridson has been recognised for his ground-breaking work on geometry, topology, and group theory in awards from the London Mathematical Society (Whitehead Prize 1999, Forder Lectureship 2005) and from the Royal Society (Wolfson Research Merit Award 2002), and by invitations to speak at the International Congress of Mathematicians in 2006 and to give the Abel Prize Lecture in Oslo in 2009.

Martin succeeds Professor Nick Woodhouse who has been President since 2012.

As someone who was drawn to mathematics and music from an early age, the connections between the two have always fascinated me. At a fundamental level the elements of music are governed by mathematics. For example, certain combinations of notes sound 'harmonious' because of the mathematical relationship between the frequencies of the notes. Musical harmony, the subdivision of music into bars and beats, the different permutations and combinations of rhythms, and so on, all give music an inherent mathematical structure. In fact just like mathematics, there is even a special notation used to describe that abstract structure. However, I think there are other, perhaps less obvious, connections. In a sense both mathematics and music are constrained, abstract, logical structures, but within these rigid constraints there is enormous freedom for creativity, with an important role played by both symmetry and beauty.

Mathematicians studying the foundations of mathematics are really studying structure, and the relationships between abstract structures. An equation $A=B$ is of course a statement of a relationship, saying that $A$ and $B$ are equivalent, in whatever sense is intended. It is straightforward enough to start writing down true equations, but this isn't what mathematicians do. Mathematicians seek interesting, elegant, or beautiful equations and structures. There is a strong aesthetic input. The way that mathematicians work, especially in the early stages of an idea, is often non-linear and intuitive, with more linear and methodical reasoning coming later. In music a composer often works in exactly the same way, but they do so for similar reasons: in both cases one is simultaneously trying to create and discover interesting and beautiful structures within a constrained system. Once you start to create, the constraints immediately lead to many consequences - sometimes wonderful consequences, but more often not what you are looking for - and one needs to use intuition to guide this simultaneous process of creation and exploration.

For some mathematicians, the connections between mathematical and musical creative processes extend further still. This was particularly true for Albert Einstein. Remarkably, he said the following about Relativity, his geometrical description of space, time and gravity: "The theory of relativity occurred to me by intuition, and music is the driving force behind this intuition. My parents had me study the violin from the time I was six. My new discovery is the result of musical perception.'' I would love to have been able to ask him more about what he meant by this! His wife Elsa once remarked: "Music helps him when he is thinking about his theories. He goes to his study, comes back, strikes a few chords on the piano, jots something down,returns to his study.'' I do the same when I'm working at home and have always regarded it as mere procrastination, but perhaps there's something deeper going on. The aesthetics one is seeking in mathematics and theoretical physics are common also in music. I think Einstein was looking for simplicity, harmony and beauty in his work, and music was for him an inspiration for this.

The notion of beauty in mathematics is hard to make precise, but for me one aspect of it has something to do with finding simplicity and complexity at the same time. By 'simple' here of course we don't mean trivial, but rather something natural and elegant; and the complexity is often initially hidden, to be uncovered by the mathematician. For example, take group theory, which is the study of symmetry in mathematics. The axioms of group theory are extremely simple, but it took hundreds of mathematicians more than a century to understand and classify the basic building blocks of these structures, which include extraordinarily complicated mathematical objects. To paraphrase the mathematician Richard Borcherds, there is no obvious hint that anything like this level of complexity exists, hidden in the initial definition. This is the sort of thing that mathematicians find beautiful. Of course, symmetries and patterns play a central role in both mathematics and music, and this is perhaps another reason why so many people are attracted to both.

The combination of simplicity, complexity, symmetry and beauty in music reaches a pinnacle in the compositions of Johann Sebastian Bach. Much of Bach's music makes use of counterpoint, where independent melodies are woven together. He often builds large, complex musical works, with many such simultaneous melodies, starting from only a small fragment of a theme. Bach then systematically works through different combinations and permutations, much like a mathematician might, making repeated use of symmetry and patterns. Writing music like this involves a great deal of analytical skill, and is very similar to solving a mathematical problem. Starting with a small, simple idea, and creating/discovering a large structure from it is very appealing to mathematicians - it is elegant. It perhaps also inspired Einstein, who was a great admirer of Bach's music. Bach's genius meant that he was able to use this approach to create beautiful music that also has a more abstract mathematical beauty. For me, it's this combination that makes his music so special.

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James Sparks and City of London Sinfonia - Bach and the Cosmos

9th October, 7.30pm-9.15pm, Mathematical Institute, Oxford, OX2 6GG

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James Sparks - Bach and the Cosmos (30 minutes)

City of London Sinfonia - J S Bach arr. Sitkovetsky, Goldberg Variations (70 minutes)

Oxford Mathematics of Consciousness and Applications Network (OMCAN) is a new network with a focus on bringing mathematics to bear on one of sciences' greatest challenges.

Over the last few decades scientists from various disciplines have started searching for the general theoretical bases of consciousness and answers to related questions such as how can consciousness be unified with physics, what medical, ethical and commercial benefits might theoretical progress bring, and is there a type of mathematical structure with the property of consciousness. This has resulted in several new mathematically formulated theories (or partial theories) of consciousness, many of which are complementary to each other. Whilst these theories are preliminary, advances in computer science are rapidly being made involving ever more parallel systems, often inspired by biological architectures, which highlights the pressing need for a step change in the level of research being undertaken to establish the general theoretical bases of consciousness.

Oxford Mathematics of Consciousness and Applications Network (OMCAN) provides researchers from across the University of Oxford with the opportunity to share their knowledge in this area, participate in relevant seminars and discussions, and find funding in support of collaborative research. Supported by the Mathematical Physical and Life Sciences Division in Oxford, it will be based at the Mathematical Institute.

OMCAN is holding its networking launch event on 19th September and you can attend and give a short introduction about yourself and your relevant interests. Please RSVP by 31 August to omcan@maths.ox.ac.uk and include up to three slides in pdf format about your relevant research interests.

Prof. Steve Furber (University of Manchester) is giving the OMCAN Inaugural Lecture on 7th November titled 'Biologically-Inspired Massively-Parallel Computation on SpiNNaker (Spiking Neural Network Architecture).

Oxford Mathematician Mehdi Yazdi talks about his study of tangled ropes in 3-dimensional space.

"A 3-manifold is a space that locally looks like the three dimensional space surrounding us. For example, imagine the complement of a closed, possibly tangled rope (i.e., knot) inside 3-dimensional space (Figure 1). One of the essential goals of topology is to understand objects, up to continuous deformations. Therefore, if one can continuously deform one knot into another one, we consider the knots and the corresponding 3-manifolds (i.e., the knot complements) to be the same.

Figure 1

A surface is a two dimensional manifold, meaning that it locally looks like a plane. Examples are a sphere, an American doughnut (i.e., torus, which has one handle) and a pretzel (doughnut with multiple handles!). If we were to remove a small open disk from each of these surfaces, we’d get what we refer to as a surface with boundary. The genus of the surface is the number of handles it has; for example, a doughnut has genus one and a sphere has genus zero.

Given two knots that a-priori look totally different, it might be the case that one can be deformed into the other. For example see the knot in Figure 2, which was considered by Ken Miller originally. It is a good exercise to try to untangle this knot (it is possible!). So, a natural question is: how can one tell two knots apart? To answer this question, topologists define invariants for knots. An invariant is a number (or another algebraic object such as a polynomial, a group, etc.) that does not vary under continuous deformations, hence the name. Therefore, if two starting knots have different invariants, they could not be the same, even up to deformations.

Figure 2

A classical and important invariant of a knot is the knot genus. Given any knot, there is an orientable surface in the 3-dimensional space, whose boundary consists of the knot, called a Seifert surface. This is illustrated in Figure 3. This surface is topologically the same as a doughnut with one disk removed from it, even though it looks different! This means that it can be deformed (possibly inside a higher dimensional space) to the standard picture of a doughnut with one disk removed. The genus of the knot K is the minimum genus between all possible orientable surfaces whose boundary is equal to K.

Figure 3

It turns out that determining the genus of a knot is a very difficult question. In fact according to the work of Agol-Hass-Thurston, if we allow both the knot and the ambient manifold to vary, then the question of determining the genus of a knot is NP-complete, roughly meaning that it is as hard as any famous computational problem in computer science. To this date, the only practical way of determining the genus of a knot uses what is called the theory of foliations. The terminology is inspired by the foliations of stratified rocks in geography.

A foliation of a 3-manifold is a decomposition of the 3-manifold into surfaces (called leaves), such that locally, the surfaces fit together like a stack of papers. The caveat is that we allow non-compact surfaces as well. As an example, see the Reeb foliations of the solid torus in Figure 4.

Figure 4

A foliation is called Reeb-less if there is no copy of the foliated solid torus as in Figure 3 anywhere in the 3-manifold. It was Thurston’s discovery that a Seifert surface S for a knot K has the smallest possible genus, if and only if S is a leaf of a Reeb-less foliation of the complement of K. This is only one of many interesting applications of Reeb-less foliations for deducing non-trivial facts about 3-manifolds. Another property being detected by Reeb-less foliations (discovered by Novikov) is whether the manifold has a non-trivial prime decomposition (which we will not define here). Therefore understanding the structure of Reeb-less foliations, and classifying them is of importance.

Kronheimer and Mrowka proved that if we consider foliations up to continuous deformation (of their tangent plane fields), then there are, typically, finitely many Reeb-less foliations on a fixed ambient 3-manifold M. This gives some hope for the classification.

By the work of Pontrjagin and Stiefel, the homotopy classes of plane fields on M are classified by two invariants. One of them is the Euler class of the tangent bundle, which lives in the second cohomology group of M. Our attention will be on the Euler class from now on.

Building on the work of Roussarie, Thurston proved that the evaluation of the Euler class of a Reeb-less foliation on a compact surface cannot exceed the Euler characteristic of the surface in absolute value. In this case,we say that the Euler class has norm at most one. In 1976 Thurston conjectured that a converse should be true when M is hyperbolic (which is a `generic’ condition by Thurston’s work on hyperbolisation of 3-manifolds). That is, every cohomology class of norm one is realised as the Euler class of some Reeb-less foliation on M.

Together with David Gabai, we disproved this conjecture. The main tool is the fully-marked surface theorem, which can have other potential applications. Recall that the evaluation of the Euler class of a Reeb-less foliation on a surface, cannot exceed the complexity of the surface. On the other hand, the equality happens for every compact leaf of the foliation. The fully-marked surface theorem gives a converse to this statement for closed hyperbolic manifolds, up to homotopy of plane fields of Reeb-less foliations."

For more on Mehdi's work together with David Gabai at Princeton University click here.

How does cellular metabolism change in different environments? Metabolism is the result of a highly enmeshed set of biochemical reactions, naturally amenable to graph-based analyses. Yet there are multiple ways to construct a graph representation from a metabolic model.

This work, a collaboration between Oxford Mathematician Mariano Beguerisse and colleagues in mathematics and bioengineering from Imperial College and the Polytecnic University of Valencia, proposes a principled framework to construct genome-scale models of metabolism using modern network science. These models resolve various challenges, such as the incorporation of pool metabolites, directionality of metabolic flows, and scenario-specific flux information.

This framework abandons metabolic descriptions that are generic blueprints in favour of tailored metabolic descriptions under any specific context of interest. For example, this model predicts the way in which the metabolism of Escherichia coli re-routes metabolic flows as environmental conditions change from being oxygen-rich (aerobic) to oxygen-poor (anaerobic). In many situations the reactions that constitute metabolic pathways form tighly-knit clusters, but in other cases the reactions may be dispersed, or not even connected to the network. Creating scenario-specific metabolic networks allows us to study how pathways behave in different conditions, and understand the role that individual reactions play. An analysis of the metabolism of human liver cells with a rare metabolic disease identifies several key reactions that flux-only analyses miss because they do not incorporate the rewiring of the metabolic network.

The method can be integrated into pipelines based on flux balance analysis and provides a systematic framework to explore changes in network connectivity as a result of environmental shifts or genetic perturbations. The paper giving more details on the research appears in the journal NPJ Systems Biology and Applications.

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The image above shows two metabolic reaction networks of E. coli. The top network shows the metabolic connectivity under normal conditions. When the oxygen is removed from the environment, the cell must drastically re-wire its metabolism in order to survive (bottom). Click to enlarge.

In many natural systems, such as the climate, the flow of fluids, but also in the motion of certain celestial objects, we observe complicated, irregular, seemingly random behaviours. These are often created by simple deterministic rules, and not by some vast complexity of the system or its inherent randomness. A typical feature of such chaotic systems is the high sensitivity of trajectories to the initial condition, which is also known in popular culture as the butterfly effect. A better understanding of chaos has been the subject of intensive mathematical research for many decades.

The time lags in non-monotone feedback may also cause chaotic behavior, a notable example being the so-called Mackey-Glass equation. This nonlinear delay differential equation was proposed by Canadian scientists Leon Glass and Michael Mackey as a prototype equation for physiological regulatory processes. Variants of this equation have been successfully used to better understand and treat various types of disorders of the hematopoietic system.

where all parameters are positive. The apparent simplicity of this model equation is rather deceiving: the Mackey-Glass equation encapsulates incredibly rich mathematical structures embedded into an infinite dimensional phase space, and has provided a lot of work for researchers over the past forty years: to date, more than 4000 papers have cited the original study, yet the emergence of its complex dynamics is not fully understood.

The reason for the presence of the time delay in the model is that following a loss of blood cells, it can take many days before new blood cells can be produced through the activation, differentiation, and proliferation of the appropriate blood stem cells. The unimodal shape of the production term is due to the fact that when the blood cell count is very high, the body does not need more blood cells hence production is low; while when the cell count is low, then your body is in bad shape and hence unable to produce enough. There is an intermediate level when the cell production runs at maximal rate. The interplay of the non-monotone feedback and the time delay makes the behaviour of solutions really tricky.

A specific area of chaos theory research is concerned with the so-called chaos control. Since chaotic behaviour is often undesirable, it is important to understand the ways in which chaos can be avoided. The traditional way of controlling a chaotic system is to pick one of the infinitely many unstable periodic solutions from the chaotic attractor, and add a small perturbation which has the effect of stabilizing this periodic solution. In a recent study, Oxford Mathematician Gergely Röst with Gábor Kiss from Szeged, Hungary, were able to control Mackey-Glass chaos with a completely different approach, which does not require a previous determination of the unstable periodic orbits of the system before the controlling algorithm is designed. The idea is to push all solutions into a domain of the phase space where the feedback is monotone, and keep them there. This way, the long term behaviour is governed by monotone delayed feedback, for which a Poincaré-Bendixson type theorem holds, ensuring that all solutions will converge either to an equilibrium or to a periodic orbit. Hence, chaotic behaviour is excluded. The researchers have shown that several popular control mechanisms can be used to control Mackey-Glass chaos, if the parameters are properly chosen. This way some previously empirically observed controls have been mathematically explained, and also new ways of control have been found such as state dependent delay control.

Oxford Mathematician Vidit Nanda talks about his and colleagues Harald Oberhauser and Ilya Chevyrev's recent work combining algebraic topology and stochastic analysis for statistical inference from complex nonlinear datasets.

"It is not difficult to generate very complicated dynamics via very simple equations. Consider, for each parameter r > 0 and natural number n, the update rules

x_{n+1} = x_n + r y_n (1-y_n) mod 1, and
y_{n+1} = y_n + r x_n(1-x_n) mod 1,

where the "mod 1" indicates that we restrict to the fractional part of the number, so for instance 3.7656 mod 1 is just 0.7656. These equations constitute a dynamical system on the unit square, and it turns out that the value of r makes an enormous difference to the behavior of this dynamical system. Below are typical pictures of the orbits (x_n,y_n) obtained (at r = 2.5, 3.5, 4, and 4.1 respectively) by applying the update rules to random initial choices of (x_0, y_0). The task for our machine, then, is to determine which r value has produced a given picture. If you were to see a fifth picture generated at one of these four r-values, you would have no trouble whatsoever determining which r was used. But it turns out to be very hard to efficiently teach a machine how to accurately make the distinction.

The difficulty lies, of course, in the nonlinearity of the dynamical system at hand. While machine learning methods are essentially linear, the geometry of the patterns is decidedly more complicated. One way to capture coarse nonlinear geometry is via the methods of topological data analysis. These reduce complicated point clouds (such as the aforementioned orbit images) to persistence barcodes, which are simply collections of intervals [b,d) labelled by geometric dimension. In our case of 2D images, the only interesting dimensions are 0 and 1. An interval [b,d) in dimension 1 indicates that when the points were thickened to balls of radius b, a hole appeared in the image, and that this hole was filled in upon thickening further to a radius d > b. Fortunately, barcodes generated at different r-values are very different while barcodes generated at the same r-value are quite similar. Unfortunately, the space of barcodes itself is nonlinear, and hence not directly amenable to machine learning.

In order to allow machine learning methods to accept barcodes as input, we linearize the space of barcodes by turning them into paths. There are several nice ways of doing this, and the picture below indicates one of them: sort the intervals in a given barcode in descending order by their length, and construct the envelope curves obtained by joining all the successive b-values and d-values so obtained. Thus, each barcode produces two paths, and one can now compute the signature of those paths to obtain a (linear!) feature map that contains all the nonlinear geometric information necessary for our parameter-inference problem. On a standard benchmark dataset, this "barcode to path to signature" managed to correctly determine the r-value with an accuracy of 98.1%."

Oxford University is committed to encouraging as wide a range of applicants as possible. Oxford Mathematics is part of that commitment. But what does that mean in practice? Well over the Summer months it means UNIQ, Oxford’s way of breaking down barriers and building bridges. A kind of construction work for the mind.

Over the last two weeks, ninety students from schools around the country have visited us in the Mathematical Institute on the UNIQ Summer Schools. These summer schools offer an impression of what it’s actually like to study Maths at Oxford. Places are given to students who are doing well at school, who are from areas of the country with low progression to university, or from low socio-economic status backgrounds. So far, so good, but what do they actually do?

Well, the week consists of taster lectures and tutorials, and, crucially, plenty of opportunities to talk about maths, both with each other and with our team of student ambassadors. Lots of the students say that meeting other people who are interested in maths is the best part of the summer school; for some of them, no-one else at their school or sixth form is as keen on maths as they are, (a refrain that persists well beyond school of course).

During the week the students have had a fascinating series of talks on topics including Benford’s Law, the Twin Paradox and the game theory of the TV show The Chase. But they have also been working together on group presentations on their favourite topics in mathematics and they’ve been working together modelling projects - open-ended problems which they’re free to approach with a variety of methods which give them an insight in to how maths actually works and enables them to spend time trying out different ideas, a luxury they may not get at school.

For example, groups have been comparing strategies to tackle malaria, investigating refraction, and optimising a bridge network. We use these projects to give the students an impression of what tutorials are like; each group has a half-hour tutorial on their project with a member of our faculty. By giving the students a first-hand experience of studying at Oxford, we can break down some of the myths, and make the whole system more transparent.

As well as giving the students a taste of the mathematics that they might study, the UNIQ summer schools also give the students a chance to experience life in Oxford. They’ve been staying in St. Anne’s College and New College, where they’ve had a quiz night, a scavenger hunt and a ghost tour, before a party on the last evening. Life in Oxford is not so different to anywhere else.

Throughout the week, the students have been helped and guided by a fantastic team of ambassadors, who are all current students or recent graduates of Oxford. One of the signs of success of the UNIQ summer schools is the high application rate to study at Oxford from UNIQ students on the summer school, and some of the ambassadors were themselves previously on UNIQ summer schools as students.

Thank you to everyone. There is much to be done, but in some not so small part of the mathematical world, progress is being made.

Each summer, a group of very enthusiastic teenage mathematicians come to spend six weeks in Oxford, working intensively on mathematics. They are participants in the PROMYS Europe programme, now in its fourth year and modelled on PROMYS in Boston, which was founded in 1989. One of the distinctive features of the PROMYS philosophy is that the students spend most of the programme discovering mathematical ideas and making connections for themselves, thereby getting a taste for life as a practising mathematician.

Mornings start with a number theory lecture followed by a problems sheet, which sounds very traditional. But at PROMYS Europe, the lectures are always at least three days later than the material comes up on the problems sheets! This allows the students to have their own mathematical adventures, exploring numerical data and seeking patterns, then proving their own conjectures before the ideas are discussed in a lecture. Another crucial part of PROMYS Europe is the community feel. This year there are 21 students participating for the first time, and six who have returned for a second experience. In addition, there are eight undergraduate counsellors, who mentor the students. Each counsellor gives daily individual feedback to their three or four students, allowing each student to progress at their own rate and to focus on their own particular interests. The counsellors are also working on their own mathematics - this year they are teaching themselves about p-adic analysis. The returning students are working in small groups on research projects, and this year are also exploring group theory. The PROMYS Europe faculty are also available to the students for much of the time, reinforcing the supportive and collaborative nature of the programme.

The occasional guest lectures give the participants glimpses of current research mathematics and of topics beyond the programme. So far, in the first two weeks of the 2018 programme students have learned about Catalan numbers and quivers from Konstanze Rietsch (King's College London), and Andrew Wiles (University of Oxford) spoke about using analysis to solve equations.

As Andrew said: "PROMYS has done very impressive work over many years in creating an environment in Boston in which young mathematicians from all over the United States can immerse themselves in serious mathematical problems over several weeks, without distraction. It is an exciting development that PROMYS and the Clay Institute have now opened up the same opportunity in Europe."

The programme is very intensive, and students spend a great deal of time grappling with challenging mathematical ideas through the daily problem sets. At the weekends, students have extra-long weekend problem sets, but also have time to explore Oxford and the surrounding area. So far this has included a tour of Oxford colleges, the chance to go punting, and a visit to Bletchley Park and the National Museum of Computing.

As in previous years, this year's group is very international, coming from 15 countries across Europe. Students have to demonstrate a sufficient command of English when they are applying, and the international language of mathematics soon transcends linguistic and cultural differences once participants arrive!

Students apply to attend PROMYS Europe, and are selected based on their mathematical potential, as displayed in their work on a number of very challenging problems. This year there were more than 200 applications for around 21 places: the students who are invited to participate have produced exceptional work on the application problems, and displayed significant commitment and mathematical maturity. The programme is dedicated to the principle that no student should be unable to attend PROMYS Europe due to financial need, and is able to provide partial and full financial aid to students who would otherwise be unable to participate.

Alumni of PROMYS in Boston have gone on to achieve at high levels in mathematics. More than 50% of PROMYS alumni go on to earn a doctorate, and 150 are currently professors, many at top universities in the US. PROMYS Europe alumni are also proving to be dedicated to pursuing mathematical studies, with several now studying at the University of Oxford. Of this year's eight counsellors, seven previously participated in PROMYS or PROMYS Europe as students, and four are Oxford undergraduates.

PROMYS Europe is a partnership of PROMYS, Wadham College and the Mathematical Institute at the University of Oxford, and the Clay Mathematics Institute. The programme is generously supported by its partners and by further financial support from alumni of the University of Oxford and Wadham College, as well as the Heilbronn Institute for Mathematical Research.