Wednesday, 2 August 2017

Landon Clay, founder of the Clay Mathematics Institute and generous supporter of Oxford Mathematics

With the passing of Landon T. Clay on 29 July, Oxford Mathematics has lost a treasured friend whose committed support and generosity were key factors in the recent development of the Mathematical Institute. The support of Landon and his wife Lavinia was the indispensible mainstay of the project to create the magnificent new home for Oxford Mathematics in the Andrew Wiles Building; the building is a symbol of the enduring legacy of their insightful, incisive support for mathematics and science. Landon's membership of the University of Oxford's Chancellor’s Court of Benefactors also recognised the breadth of his support for many parts of the University, always with a sharp emphasis on supporting excellence.

Landon Clay was the Founder of the Clay Mathematics Institute, which has had a profoundly beneficial effect on the progress and appreciation of research into fundamental mathematics. He will perhaps be best remembered for his inspired creation of the Millennium Prizes: these have the crucial feature that they draw the public’s attention to the fundamental importance of the prize problems themselves, in contrast to the focus on the prize-winners as is the case with the other great prizes of mathematics.

The Clay Mathematics Institute, directed from the President’s Office in the Andrew Wiles Building, supports mathematical excellence in many other ways. In particular, the Clay Research Fellowships give the brightest young mathematicians in the world five years of freedom to develop their ideas free of financial concerns and institutional demands. The fruits of this programme can be implied from the fact that three of the four Fields Medallists at the International Congress in 2014 were former Clay Fellows.

The ramifications of Landon Clay’s generous and astutely directed support for mathematics will echo long into the future. A fuller account of his life and the range of his philanthropy can be found on the Clay Mathematics Institute website.

Photograph by Robert Schoen, 2004

Friday, 28 July 2017

Knots and the nature of 3-dimensional space

It is an intriguing fact that the 3-dimensional world in which we live is, from a mathematical point of view, rather special. Dimension 3 is very different from dimension 4 and these both have very different theories from that of dimensions 5 and above. The study of space in dimensions 2, 3 and 4 is the field of low-dimensional topology, the research area of Oxford Mathematician Marc Lackenby.

One of the reasons that 3-dimensional space is different from the others is the presence of knots. A knot is just a piece of string that is usually closed up to form a loop (mathematically, it is a smoothly embedded simple closed curve). It is a familiar everyday fact that there are many different knots, the simplest two being the unknot and the trefoil shown below. However, if you put a knotted piece of string into 4-dimensional space, you can always unknot it.


The existence of non-trivial knots is a key feature of 3-dimensional space, and so it is a worthwhile goal to attempt to classify knots. One is immediately led to the following simple questions: given two knot diagrams, how can we decide whether they are the same knot? In fact, how can we even decide whether a knot diagram represents the unknot? These questions are simple to state, but actually are very difficult to answer. What is needed is an algorithm that can definitively resolve such questions in finite time. It is known that similar problems in high dimensions are unsolvable, but the situation in dimension 3 is tractable, just.

It is an old theorem (dating back to the 1920s) that any two diagrams of a knot differ by a sequence of Reidemeister moves, which are local modifications to a diagram, shown below:

This has the following algorithmic consequence: if two diagrams represent the same knot, then it will always be possible to prove this, as follows. Apply all possible Reidemeister moves to one of the diagrams. Then apply all possible Reidemeister moves to each of the resulting collection of diagrams, and so on. If the two knots are the same, this procedure will eventually reach the second diagram and so you will have proved that the two knots are equivalent. But if the knots are different, this process will not terminate. So, to turn this into an effective algorithm to decide whether two knots are the same, one needs to be given, in advance, an upper bound on the number of Reidemeister moves required to relate two diagrams of a knot. The search for such a bound is what Marc Lackenby has been working on recently. He has shown that for any diagram of the unknot with c crossings, there is a sequence of at most $(236\ c)^{11}$ moves that takes it to the diagram with no crossings. The bound $(236\ c)^{11}$ may seem large, but it is actually much smaller than what was known previously, which was an exponential function of c. The existence of such a polynomial bound had been a well-known longstanding problem. To prove this theorem, Marc had to use a wide variety of different techniques from across low-dimensional topology. His paper was recently published in the Annals of Mathematics.

This polynomial bound is not the end of the story. The procedure for deciding whether a knot is the unknot using Reidemeister moves is simple but not particularly efficient. Even with the polynomial bound on the number of moves, the running time of the algorithm is an exponential function of the initial crossing number c. Can one do better than this? No-one knows, but Marc is currently working on this problem, and hopes to find an algorithm that runs in sub-exponential time.

Friday, 28 July 2017

Numerical Analyst Nick Trefethen on the pleasures and significance of his subject

Oxford Mathematician Nick Trefethen was recently awarded the George Pólya Prize for Mathematical Exposition by the Society for Industrial and Applied Mathematics (SIAM) "for the exceptionally well-expressed accumulated insights found in his books, papers, essays, and talks." Here Nick refllects on the award, his approach to mathematics and the ever-expanding role of Numercial Analysis in the world.

Congratulations on your award, how did you react when you found out you had won?

I was thrilled. There are many accolades to dream of achieving in an academic career but I am one of the relatively few mathematicians who love to write. So, to be acknowledged for mathematical exposition is important to me. My mother was a writer and I guess it is in my blood.

What is Numerical Analysis?

Much of science and engineering involves solving problems in mathematics, but these can rarely be solved on paper. They have to be solved with a computer, and to do this you need algorithms. 

Numerical Analysis is the field devoted to developing those algorithms.  Its applications are everywhere. For example, weather forecasting and climate modelling, designing airplanes or power plants, creating new materials, studying biological populations, it is simply everywhere.

It is the hands-on exploratory way to do mathematics. I like to think of it as the fastest laboratory discipline. I can conceive an experiment and in the next 10 minutes, I can run it. You get the joy of being a scientist without the months of work setting up the experiment.

How does it work in practice?

Everything I do is exploratory through a computer and focused around solving problems such as differential equations, while still addressing basic issues. In my forthcoming book Exploring ODEs (Ordinary Differential Equations) for example, every concept measured is illustrated as you go using our automated software system, Chebfun.

How has your research advanced the field?

Most of my own research is not directly tied to applications, more to the development of fundamental algorithms and software.

But, I have been involved in two key physical applications in my career. One was in connection with transition to turbulence of fluid flows, such as flow in a pipe; and recently in explaining how a Faraday cage works, such as the screen on your microwave oven that keeps the microwaves inside the device, while letting the light escape so that you can keep an eye on your food.

You got a lot of attention for your alternative Body Mass Index (BMI) formula, how did you come up with it?

My alternative BMI formula was not based on scientific research. But, then again, the original BMI formula wasn’t based on much research either. I actually wrote a letter to The Economist with my theory. They published it and it spread through the media amazingly.

As a mathematician, unless you’re Professor Andrew Wiles or Stephen Hawking for example, you are fortunate to have the opportunity to be well known within the field and invisible to the general public at the same time. The BMI interest was all very uncomfortable and unexpected.

Why do you think so few mathematicians are strong communicators?

I don’t think this is necessarily the case. One of the reasons that British universities are so strong academically, is the Research Excellence Framework, through which contributions are measured. But, on the other hand the structure has exacerbated the myth that writing books is a waste of time for academic scientists. The irony is that in any real sense, writing books is what gives you longevity and impact.

At the last REF the two things that mattered most to me, that I felt had had the most impact, were my latest book and my software project, and neither were mentioned.

In academia we play a very conservative game and try to only talk about our latest research paper. The things that actually give you impact are not always measured.

What are you working on at the moment?

I just finished writing my latest book on ODEs (due to be published later this year), which I am very excited about.

Have you always had a passion for mathematics?

My father was an engineer and I sometimes think of myself as one too - or perhaps a physicist doing maths. Numerical Analysis is a combination of mathematics and computer science, so your motivations are slightly different. Like so many in my field, I have studied and held faculty positions in both areas.

What is next for you?

I am due to start a sabbatical in Lyon, France later this year. I'll be working on a new project, but if you don’t mind, I won’t go into detail. A lot of people say that they are driven by solving a certain applied problem, but I am really a curiosity-driven mathematician. I am driven by the way the field and the algorithms are moving. I am going to try and take the next step in a particular area. I just need to work on my French.

What do you think can be done to support public engagement with mathematics?

I think the change may come through technology, almost by accident. You will have noticed over the last few decades, that people have naturally become more comfortable with computers, and I think that may expand in other interesting directions.

The public’s love/hate relationship with mathematics has been pervasive throughout my career.  As a Professor, whenever you get to border control you get asked about your title. ‘What are you a Professor of?’ When you reply, the general response is ‘oh I hated maths.’ But, sometimes you'll get ‘I loved maths, it was my best subject’, which is heartening.

What has been your career highlight to date?

Coming to Oxford was a big deal, as was being elected to the Royal Society. It meant a lot to me, especially because I am an American. It represented being accepted by my new country.

Are there any research problems that you wish you had solved first?

I’m actually going to a conference in California, where 60 people will try to prove a particular theorem; Crouzeix’s Conjecture. By the end of the week I will probably be kicking myself that I wasn’t the guy to find the final piece of the puzzle.

Friday, 28 July 2017

Oxford Mathematics Research: Nikolay Nikolov on his latest research into Sofic Groups

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Nikolay Nikolov discusses his research in to Sofic Groups.

"In the first year of mathematics at Oxford we learn Cayley's theorem that every finite group is isomorphic to a subgroup of the symmetric group $S_n$ for some integer $n$. Many problems in group theory are motivated by analogues of Cayley's theorem where we want to approximate a general infinite group $\Gamma$ by permutations. One far reaching generalization of such approximations is the notion of sofic group. Roughly speaking instead of a homomorphism from $\Gamma$ into $S_n$ we insist that for any $\epsilon >0$ there is a map from $\Gamma$ to $S_n$ which is '$\epsilon$-close' to being a homomorphism. One slick way to define this is using the notion of ultrafilters and ultralimits from logic. For the definition of ultrafilters and ultralimits you can consult

For a permutation $\pi \in S_n$ let us denote by $r_n (\pi)= \frac{\mathrm{mov} (\pi) }{n}$ where $\mathrm{mov} (\pi)$ is the number of points moved by $\pi$. The distance function $d_n(x,y)= r_n (x^{-1}y)$ is called the Hamming metric on $S_n$. For a nonprincipal ultrafilter $\omega$ on $\mathbb N$, let $M_\omega$ be the metric ultraproduct of the groups $S_n$ with respect to their metrics $d_n$. More precisely $M_\omega$ is the quotient group $\frac{G}{K_\omega}$ where $G=\prod_{n=1}^\infty S_n$ and $K_\omega$ is the set of sequences $(\pi_n)_n$ with $\pi_n \in S_n$, such that the ultralimit of the sequence $r_n(\pi_n)$ is $0$. As an exercise you can check that $K_\omega$ is indeed a normal subgroup of $G:=\prod_n S_n$.

A countable group $\Gamma$ is sofic if $\Gamma$ is isomorphic to a subgroup of $M_\omega$. It turns out that this definition does not depend on the choice of $\omega$.

It is a major open question in group theory whether every group is sofic. If true this will imply several other conjectures, for example Kaplansky's Direct Finiteness Conjecture: If $\Gamma$ is a group, $K$ is a field and $a,b$ are two elements of the group ring $K[\Gamma]$ such that $ab=1$, then $ba=1$. Many groups are known to be sofic, for example abelian groups, solvable groups and linear groups (subgroups of $GL(m,K)$ for a field $K$).

We don't know if every group is sofic but we know a little about the groups $M_\omega$: these are simple uncountable groups. In fact the $M_\omega$ together with $C_2$ are all the simple quotients of $G$. The group $G$ with the product topology is an example of a compact Hausdorff group, like the circle $S^1$ and its generalizations the unitary groups $U(m)$. One difference which sets $G$ apart from $U(m)$ is that while unitary groups are connected, our $G$ is totally disconnected (i.e. its connected components are singletons), in fact $G$ is topologically homeomorphic to the Cantor set). A compact Hausdorff group with this property is called a profinite group.

So in order to find out whether every group is sofic we first need to know about the quotients of compact groups. One step in this direction was taken by me and Dan Segal where we proved the following theorem: a finitely generated quotient of a compact Hausdorff group must be finite. If in addition the compact group is connected then one can deduce that the quotient must be in fact the trivial group. Dan Segal and I also showed that the presence of abelian groups is responsible for the existence of countably infinite quotients of compact groups. For example the circle $S^1$ does not have a finite quotient but has a countably infinite quotient (Exercise: prove this!). The same is true for any infinite abelian compact group.

Some other recent results on sofic groups can be found here."

Wednesday, 19 July 2017

NIck Trefethen wins George Pólya Prize for Mathematical Exposition

Ocford Mathematician Nick Trefethen FRS has been awarded the George Pólya Prize for Mathematical Exposition by the Society for Industrial and Applied Mathematics (SIAM) "for the exceptionally well-expressed accumulated insights found in his books, papers, essays, and talks... His enthusiastic approach to his subject, his leadership, and his delight at the enlightenment achieved are unique and inspirational, motivating others to learn and do applied mathematics through the practical combination of deep analysis and algorithmic dexterity."

Nick is Professor of Numerical Analysis and Head of the Numerical Analysis Group here in Oxford. 

Monday, 10 July 2017

Oxford Mathematicians win outstanding certificate as part of the new IIF Tao Hong Award

Oxford Mathematicians Stephen Haben and Peter Grindrod and colleagues have won an outstanding certificate as part of the new IIF Tao Hong Award for papers in energy forecasting published in the International Journal of Forecasting.

The paper, 'A new error measure for forecasts of household-level, high resolution electrical energy consumption,' provides high-quality verification tools for load forecasts, which are essential in managing power systems. This is particularly helpful for work on demand profiling in the residential sector, where the temporal resolution of data has increased rapidly in recent years.

Monday, 10 July 2017

Shapes and Numbers - Oxford Mathematics Research considers number theory and topology

As part of our series of research articles deliberately focusing on the rigour and intricacies of mathematics, we look at Oxford Mathematician Minyhong Kim's research in to the relationship between number theory and topology. Minhyong Kim is Professor of Number Theory here in Oxford and Fellow of Merton College.

It is probably well-known that number theory is the source of some of the oldest and most accessible questions in mathematics:

Which regular polygons can be constructed using only a straight-edge and a compass?

How are the primes distributed in the large?

What are the integral solutions of the equation $x^n+y^n=z^n$?

Is there an algorithm to generate the rational solutions to rational cubic equations of the form $y^2=x^3+ax+b$?

Many people with mathematical inclinations will be drawn to the natural simplicity of these queries, perhaps early on in life, and will gradually expand the reservoir of their knowledge in the hope of approaching a solution, progressing from the easy to the hard cases. Mathematics as a whole has had a historical relationship to number theory parallel to the personal development of a mathematician. Over millennia, logic, algebra, analysis, and geometry have all been employed in the service of number theory, the tools of the trade often becoming more powerful or refined in response to the demands of arithmetic problems. In our times, it's difficult to find a single area of mathematics, from the elementary to the most conceptually sophisticated, which is not used in some serious way in number theory. Number theory has acquired thereby the status of a quite general testing ground for conceptual progress: a mysterious and abstract theory can prove its worth and its connection to mathematical reality by its applicability to concrete problems of number theory. The tendency to generate natural problems has enabled number theory to reconstitute itself as a pure laboratory for the power of ideas.

Since the 1960s, a major instance of this interaction has been that between topology and number theory. Topologists are adept at coming up with extremely abstruse notions of shape and space, such as a topos, a simplicial set, or a spectrum. Each of these in turn have had fruitful number-theoretic incarnations in the study of equations over finite fields, quadratic forms, and in the Galois theory of $p$-adic fields. Minhyong Kim's research continues the exploration of the relationship between topology and number theory, most actively with ideas from homotopy theory. As an elementary example, consider a set of paths $P(1 ,z)$ in the punctured complex plane from point $1$ to the point $z$. Surprisingly, there is a big difference in structure between the path spaces with $z$ transcendental and those with $z$ algebraic. More precisely, in the latter case, the homotopy classes of paths, suitably completed, admit an intricate symmetry group coming from Galois theory. Kim has been studying the classification of such hidden symmetry with a view to detecting rational or algebraic solutions to polynomial equations. For example, using only the Galois symmetries of the path spaces between solution points, he has been able to reprove the theorem, first proved by Faltings then vastly improved by Wiles, that the equation $$x^n+y^n=1$$ has only finitely many rational solutions when $n\geq 4$ (this is joint work with John Coates).

Most recently, in collaboration with many mathematicians and physicists from the US, Europe, and Asia, Kim is involved in a programme to develop the ideas of topological quantum field theory in the realm of number theory. The main mathematical impact of topological quantum field theory arises from the use of an integral over fields (in the sense of the physicist, not that of the algebraist) to measure the quantum correlation between points on a space, the winding of knots, or to define a numerical invariant of the space itself. The current project explores the application of this idea to primes and rings of integers in number fields (now in the sense of algebraists), where the role of the physical field is played by representations of arithmetic Galois groups. These representations themselves lie at the crossroads of the most important paths of investigation in present day number theory, such as the conjecture of Birch and Swinnerton-Dyer, the theory of zeta and L-functions, and the Langlands programme. One goal of this research is to understand their centrality from a point of view consistent with the intuition of geometry, topology, and physics.

Friday, 7 July 2017

The Law of the Few - Sanjeev Goyal's Oxford Mathematics Public Lecture now online

The study of networks offers a fruitful approach to understanding human behaviour. Sanjeev Goyal is one of its pioneers. In this lecture Sanjeev presents a puzzle:

In social communities, the vast majority of individuals get their information from a very small subset of the group – the influencers, connectors, and opinion leaders. But empirical research suggests that there are only minor differences between the influencers and the others. Using mathematical modelling of individual activity and networking and experiments with human subjects, Sanjeev helps explain the puzzle and the economic trade-offs it contains.

Professor Sanjeev Goyal FBA is the Chair of the Economics Faculty at the University of Cambridge and was the founding Director of the Cambridge-INET Institute.




Thursday, 6 July 2017

Alex Wilkie and Alison Etheridge win LMS Prizes

Congratulations to the Oxford Mathematicians who have just been awarded LMS prizes. Alex Wilkie receives the Pólya Prize for his profound contributions to model theory and to its connections with real analytic geometry and Alison Etheridge receives the Senior Anne Bennett Prize in recognition of her outstanding research on measure-valued stochastic processes and applications to population biology; and for her impressive leadership and service to the profession.

Thursday, 29 June 2017

Oxford Mathematics Visiting Professor Michael Duff awarded the Paul Dirac Medal and Prize

Professor Michael Duff of Imperial College London and Visiting Professor here in the Mathematical Institute in Oxford has been awarded the Dirac Medal and Prize for 2017 by the Institute of Physics for “sustained groundbreaking contributions to theoretical physics including the discovery of Weyl anomalies, for having pioneered Kaluza-Klein supergravity, and for recognising that superstrings in 10 dimensions are merely a special case of p-branes in an 11-dimensional M-theory.”

Michael Duff holds a Leverhulme Emeritus Fellowship, is a Fellow of the Royal Society, the American Physical Society and the Institute of Physics and was awarded the 2004 Meeting Gold Medal, El Colegio Nacional, Mexico.