News

Wednesday, 3 October 2018

Noisy brains - Fast white noise generation for modelling uncertainty in the fluid dynamics of the brain

Over the last few years, the study of the physiological mechanisms governing the movement of fluids in the brain (referred to as the brain waterscape) has gained prominence. The reason? Anomalies in the brain fluid dynamics are related to diseases such as Alzheimer's disease, other forms of dementia and hydrocephalus. Understanding how the brain waterscape works can help discover how these diseases develop. Unfortunately, experimenting with the human brain in vivo is extremely difficult and the subject is still poorly understood. As a result, this topic is a highly active interdisciplinary research area. In our research team, formed by Oxford Mathematicians Matteo Croci, Patrick Farrell and Mike Giles in collaboration with Marie E. Rognes, Vegard Vinje and colleagues from Simula Research Laboratory, we contribute by simulating the brain waterscape on a computer, hence providing an alternative avenue of investigation that is cheap and does not require human experimentation.

One of the main challenges in brain simulation is the lack of accurate quantitative information on the mechanical input parameters needed to set up our mathematical model. Quantities such as brain matter permeability, interstitial fluid flow velocity and diffusivity are only known approximately or on average. The position of the blood vessels and capillaries can be measured, but it varies from patient to patient and it is extremely difficult to resolve without significant expense.

To overcome these issues, we can construct surrogates for these quantities that account for the uncertainty in their values through the use of Gaussian random fields. Gaussian random fields are functions of space whose values at each point are given by Gaussian random variables which are correlated according to a given covariance function. Sampling realisations of these fields can be extremely expensive computationally. Our team of researchers developed one of the fastest currently available sampling algorithms for Mat\'ern-Gaussian fields. This is achieved by recasting the Gaussian field sampling problem as the solution of an elliptic partial differential equation (PDE) driven by spatial white noise. Solving this equation is standard, but sampling white noise is not. The developed algorithm is able to draw white noise realisations quickly and efficiently and can be used in conjunction with the multilevel Monte Carlo (MLMC) method for further acceleration (see here). The coupling of white noise required by MLMC is enforced even in the non-nested case thanks to a supermesh construction. Furthermore, the algorithm can be employed to solve a wider class of PDE problems with spatial white noise additive terms.

For further information on the efficient sampling and coupling of white noise for MLMC with non-nested meshes see here. A paper on uncertainty quantification in brain simulation is in preparation and a link will appear here in due course.
 

Tuesday, 2 October 2018

The future of Mathematics - welcome to our new DPhil and Masters students

Amid all the debate about equipping ourselves for the 'technological' world of the future, one thing is for sure: the quality of research in the Mathematical and Life Sciences (and beyond) depends on the quality of its young researchers. In that spirit we are delighted to welcome our latest cohort of DPhil (PhD) students, 43 of them, all fully funded, from across the globe. 13 from the UK, 15 from the European Union and a further 15 from India, Kenya, Norway, Australia, Mexico, USA, China, Switzerland, Argentina, Israel and South Africa.  

We would also like to welcome our masters students and in particular our first cohort of Oxford Masters in Mathematical Sciences (OMMS) students, 36 in total, 26 men and 10 women from 17 different countries. This standalone course offers students the opportunity to join our current fourth year undergraduates and work with our internationally renowned faculty.

Welcome to everyone. We (and we don't just mean Oxford) need you all.

Monday, 1 October 2018

Roger Penrose's Oxford Mathematics Public Lecture, 'Eschermatics' now online

Roger Penrose's relationship with the artist M.C. Escher was not just one of mutual admiration. Roger's thinking was consistently influenced by Escher, from the famous Penrose tiling to his groundbreaking work in cosmology. The respect was mutual, as was clear when Roger dropped in to see Escher at his home...

Oxford Mathematics hosted this special event in its Public Lecture series during the conference to celebrate the 20th Anniversary of the foundation of the Clay Mathematics Institute. 

 

 

 

 

 

 

 

Thursday, 27 September 2018

Connecting the Dots with Pick's Theorem

Oxford Mathematician Kristian Kiradjiev has won the Graham Hoare Prize (awarded by the Institute of Mathematics and its Applications) for his article "Connecting the Dots with Pick's Theorem". The Graham Hoare Prize is awarded annually to Early Career Mathematicians for a brilliant Mathematics Today article. Kristian also won the award in 2017. Here he talks about his work.

"Pick's theorem is an example of a theorem that is not widely known but has surprising applications to various mathematical problems. At its essence, Pick's theorem is a geometrical result, but has quite a few algebraic implications.

Published in 1899 by Georg Alexander Pick, the theorem states that given any simple polygon whose vertices lie on an integer grid, its area, $A$, is calculated according to the following formula: \begin{equation} A=i+\frac{b}{2}-1, \label{eq:Pick} \end{equation} where $i$ is the number of grid points inside the polygon, $b$ is the number of grid points that lie on the boundary of the polygon, and by simple we mean a polygon without holes. For a polygon with $n$ holes, the formula generalises to \begin{equation} A=i+\frac{b}{2}-1+n. \label{eq:Pick1} \end{equation} A useful property of Pick's formula that we immediately note is that it is invariant under shearing of the lattice. Also, scaling the distance between grid points in one direction simply scales the area of the polygon. These two observations can be used to show that Pick's formula is valid for sheared (triangular) grids as well, provided we take account of any scaling in the direction perpendicular to the shearing one.

Pick's theorem can be used to tackle a number of problems in different fields of mathematics. To give a flavour of this, we present a geometric problem that is simple to state. Although square integer grids are ubiquitous in our life, it is a fact that one cannot draw one of the simplest figures, namely, an equilateral triangle with its vertices being grid points on such a lattice. Some standard proofs involve tedious algebra and trigonometry, whereas Pick's theorem proves the result immediately. Suppose we have drawn an equilateral triangle with sides of length $d$ on a square grid. Then, its area is given by the well-known formula $A=\sqrt{3} d^2/4$. Since the vertices of the triangle are integer points, then $d^2$ is an integer (by Pythagoras' theorem, for example), and, thus, the area is an irrational number. However, Pick's formula on square grids always gives a rational area. This contradiction proves the initial claim. The same argument shows that regular hexagons cannot be drawn on a square integer grid either. Another problem where Pick's theorem can be applied concerns some properties of the so-called Farey sequences.

One might think that Pick's theorem can be easily generalised to three (or more) dimensions for volume of solids, etc. However, in 1957, J. Reeve produced an example of what is now known as the Reeve tetrahedron, which shows that Pick's theorem does not have a direct analogue in three or more dimensions by devising a figure that can take many different values for its volume without changing the number of interior and boundary points. However, Pick's theorem has some close analogues in more than two dimensions such as the Ehrhart polynomials.

The figure above is a rather complicated polygon, called a Farey sunburst (because of its relation to Farey sequences). Its area can be readily calculated using Pick's theorem to be 48 square units."

Wednesday, 26 September 2018

Helen Byrne and Francis Woodhouse win Society for Mathematical Biology awards

The Society for Mathematical Biology has announced its 2018 Awards for established biologists and among the winners are Oxford Mathematicians Helen Byrne and Francis Woodhouse.

Helen will be the recipient of the Leah Edelstein-Keshet Prize for her work focused on the development and analysis of mathematical and computational models that describe biomedical systems, with particular application to the growth and treatment of solid tumors, wound healing and tissue engineering. This award recognizes an established scientist with a demonstrated track record of exceptional scientific contributions to mathematical biology and/or has effectively developed mathematical models impacting biology. "Dr. Byrne has made outstanding scientific achievements coupled with her record of active leadership in mentoring scientific careers." The Edelstein-Keshet Prize consists of a cash prize of $500 and a certificate given to the recipient. The winner is expected to give a talk at the Annual Meeting of the Society for Mathematical Biology in Montreal in 2019.

Francis has won the H. D. Landahl Mathematical Biophysics Award. This award recognizes the scientific contributions made by a postdoctoral fellow who is making exceptional scientific contributions to mathematical biology. The award is acknowledged with a certificate, and a cash prize of USD $500.

 

Friday, 21 September 2018

Oxford Mathematics and the Clay Mathematics Institute Public Lectures: Roger Penrose - Eschermatics WATCH LIVE MONDAY 24th SEPT 5.30PM BST

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.

You can watch live at:

https://www.facebook.com/OxfordMathematics/

OR

https://livestream.com/oxuni/Penrose

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

To whet your appetite here is Roger demonstrating the Impossible Triangle from his 2015 lecture.

Monday, 17 September 2018

Martin Bridson appointed President of the Clay Mathematics Institute

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.

Monday, 27 August 2018

Bach and the Cosmos - James Sparks previews his upcoming Oxford Mathematics Public Lecture with City of London Sinfonia, 9 October

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)

Alexandra Wood - Director/Violin

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Please email external-relations@maths.ox.ac.uk to register

Watch live:
https://www.facebook.com/OxfordMathematics
https://www.livestream.com/oxuni/Bach-Cosmos

The Oxford Mathematics Public Lectures are generously supported by XTX Markets

Wednesday, 15 August 2018

OMCAN: A new network for mathematical approaches to consciousness research

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).

 

Tuesday, 14 August 2018

From knots to foliations - understanding the complexity of tangled ropes

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.

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