We have two contrasting Oxford Mathematics Public Lectures coming up in the next ten days. One features a genius from the eighteenth century whose work is still pertinent today. The other is very much from the 21st century and illuminates the direction mathematics is currently travelling. Please email external-relations@maths.ox.ac.uk to register or follow our twitter account for details on how to watch live.

Euler’s pioneering equation: ‘the most beautiful theorem in mathematics’ - Robin Wilson. 28 February, 2018, 5-6pm

Can Mathematics Understand the Brain? - Alain Goriely, March 8th, 5.15-6.15pm

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Euler’s pioneering equation: ‘the most beautiful theorem in mathematics’ - Robin Wilson

Euler’s equation, the ‘most beautiful equation in mathematics’, startlingly connects the five most important constants in the subject: 1, 0, π, e and i. Central to both mathematics and physics, it has also featured in a criminal court case and on a postage stamp, and has appeared twice in The Simpsons. So what is this equation – and why is it pioneering?

Robin Wilson is an Emeritus Professor of Pure Mathematics at the Open University, Emeritus Professor of Geometry at Gresham College, London, and a former Fellow of Keble College, Oxford.

Can Mathematics Understand the Brain? - Alain Goriely

The human brain is the object of the ultimate intellectual egocentrism. It is also a source of endless scientific problems and an organ of such complexity that it is not clear that a mathematical approach is even possible, despite many attempts.

In this talk Alain will use the brain to showcase how applied mathematics thrives on such challenges. Through mathematical modelling, we will see how we can gain insight into how the brain acquires its convoluted shape and what happens during trauma. We will also consider the dramatic but fascinating progression of neuro-degenerative diseases, and, eventually, hope to learn a bit about who we are before it is too late.

Alain Goriely is Professor of Mathematical Modelling, University of Oxford and author of 'Applied Mathematics: A Very Short Introduction.'

Oxford Mathematician Katherine Staden provides a fascinating snapshot of the field of combinatorics, and in particular extremal combinatorics, and the progress that she and her collaborators are making in answering one of its central questions posed by Paul Erdős over sixty years ago.

"Combinatorics is the study of combinatorial structures such as graphs (also called networks), set systems and permutations. A graph is an encoding of relations between objects, so many practical problems can be represented in graph theoretic terms; graphs and their mathematical properties have therefore been very useful in the sciences, linguistics and sociology. But mathematicians are generally concerned with theoretical questions about graphs, which are fascinating objects for their own sake. One of the attractions of combinatorics is the fact that many of its central problems have simple and elegant formulations, requiring only a few basic definitions to be understood. In contrast, the solutions to these problems can require deep insight and the development of novel tools.

A graph $G$ is a collection $V$ of vertices together with a collection $E$ of edges. An edge consists of two vertices. We can represent $G$ graphically by drawing the vertices as points in the plane and drawing a (straight) line between vertices $x$ and $y$ if $x,y$ is an edge.

Extremal graph theory concerns itself with how big or small a graph can be, given that it satisfies certain restrictions. Perhaps the first theorem in this area is due to W. Mantel from 1907, concerning triangles in graphs. A triangle is what you expect it to be: three vertices $x,y,z$ such that every pair $x,y$ and $y,z$ and $z,x$ is an edge. Consider a graph which has some number $n$ of vertices, and these are split into two sets $A$ and $B$ of size $\lfloor n/2\rfloor$, $\lceil n/2\rceil$ respectively. Now add every edge with one vertex in $A$ and one vertex in $B$. This graph, which we call $T_2(n)$, has $|A||B|=\lfloor n^2/4\rfloor$ edges. Also, it does not contain any triangles, because at least two of its vertices would have to both lie in $A$ or in $B$, and there is no edge between such pairs. Mantel proved that if any graph other than $T_2(n)$ has $n$ vertices and at least $\lfloor n^2/4\rfloor$ edges, it must contain a triangle. In other words, $T_2(n)$ is the unique `largest' triangle-free graph on $n$ vertices.

Following generalisations by P. Turán and H. Rademacher in the 1940s, Hungarian mathematician Paul Erdős thought about quantitatively extending Mantel's theorem in the 1950s. He asked the following: among all graphs with $n$ vertices and some number $e$ of edges, which one has the fewest triangles? Call this quantity $t(n,e)$. (One can also think about graphs with the most triangles, but this turns out to be less interesting).

Astoundingly, this seemingly simple question has yet to be fully resolved, 60 years later. Still, in every intervening decade, progress has been made, by Erdős, Goodman, Moon-Moser, Nordhaus-Stewart, Bollobás, Lovász-Simonovits, Fisher and others. Finally, in 2008, Russian mathematician A. Razborov managed to solve the problem asymptotically (meaning to find an approximation $g(e/\binom{n}{2})$ to $t(n,e)$ which is arbitrarily accurate as $n$ gets larger). Razborov showed that, for large $n$, $g(e/\binom{n}{2})$ has a scalloped shape: it is concave between the special points $\frac{1}{2}\binom{n}{2}, \frac{2}{3}\binom{n}{2}, \frac{3}{4}\binom{n}{2}, \ldots$. His solution required him to develop the new method of flag algebras, part of the emerging area of graph limits, which has led to the solution of many longstanding problems in extremal combinatorics.

The remaining piece of the puzzle was to obtain an exact (rather than asymptotic) solution. In recent work with Hong Liu and Oleg Pikhurko at the University of Warwick, I addressed a conjecture of Lovász and Simonovits, the solution of which would answer Erdős's question in a strong form. The conjecture put forward a certain family of $n$-vertex, $e$-edge graphs which are extremal, in the sense that they should each contain the fewest triangles. So in general there is more than one such graph, one aspect which makes the problem hard. Building on ideas of Razborov and Pikhurko-Razborov, we were able to solve the conjecture whenever $e/\binom{n}{2}$ is bounded away from $1$; in other words, as long as $e$ is not too close to its maximum possible value $\binom{n}{2}$.

Our proof spans almost 100 pages and (in contrast to Razborov's analytic proof) is combinatorial in nature, involving a type of stability argument. It would be extremely interesting to close the gap left by our work and thus fully answer Erdős's question."

Revolving captions:

The graph $T_2(n)$, which is the unique largest triangle-free graph on $n$ vertices.

The minimum number of triangles $t(n,e)$ in an $n$-vertex $e$-edge graph plotted against $e/\binom{n}{2}$. This was proved in the pioneering work of A. Razborov.

Making new graphs from old: an illustration of a step in the proof of the exact result by Liu-Pikhurko-Staden.

What does boiling water have in common with magnets and the horizon of black holes? They are all described by conformal field theories (CFTs)! We are used to physical systems that are invariant under translations and rotations. Imagine a system which is also invariant under scale transformations. Such a system is described by a conformal field theory. Remarkably, many physical systems admit such a description and conformal field theory is ubiquitous in our current theoretical understanding of nature.

Two-dimensional CFTs are very special: in this case the conformal group is actually infinitely dimensional! This has led to remarkable progress over the last four decades. In contrast, CFTs in higher dimensions are notoriously difficult to deal with. First of all, in this case symmetries are much less powerful. Furthermore, the standard textbook methods to compute observables in physical theories, called perturbative methods, only apply to theories with small coupling constants (giving an expansion in powers of these couplings), but generic CFTs do not have small parameters! Because of these reasons, progress in higher dimensional CFTs proved to be much harder than in the two-dimensional case.

The breakthrough came in 2008 when it was shown how to attack higher dimensional CFTs with the conformal bootstrap program. The basic idea is to resort to symmetries and consistency conditions and ask the question: which values of the observables I am interested in would lead to a consistent CFT? The basic observables in a CFT are correlators of local operators $\langle {\cal O}(x_1) \cdots {\cal O}(x_n) \rangle$, and one of the consistency conditions is that these correlators have the right properties under permutation of the insertion points, for instance: $$\langle {\cal O}(x_1){\cal O}(x_2){\cal O}(x_3){\cal O}(x_4)\rangle =\langle {\cal O}(x_3){\cal O}(x_2){\cal O}(x_1){\cal O}(x_4)\rangle$$ Believe it or not, when combined with conformal symmetry these conditions are incredibly constraining! The conformal bootstrap proved to be very powerful, and its range of applicability is very vast: it basically applies to every CFT.

While the original proposal, and most of the work which followed it, was numeric in nature, my collaborators and I have been developing a method to obtain analytic results in CFTs, following the bootstrap philosophy. Although still in its infancy, this method has already produced remarkable results for vast families of CFTs, including both perturbative and non-perturbative theories. For perturbative theories it offers an alternative to standard diagrammatic methods, but without many of the problems associated with them. An important conformal theory that has been studied for many decades, relevant for boiling water and a generalisation of the Ising Model, is the Wilson-Fisher model. For this model, and for certain families of observables, the results from our methods have already surpassed the available results from diagrammatic techniques! It will be very exciting to see where all this leads.

Prof. Ursula Martin and Dr Ian Griffiths have each been awarded an MPLS Impact Award for 2017-18. The MPLS (Mathematical, Physical, Engineering and Life Sciences Division at the University of Oxford) Impact Awards scheme aims to foster and raise awareness of impact by rewarding it at a local level.

Ursula's award is for Public Engagement in connection with the 2015 celebrations of the 200th anniversary of Ada Lovelace's birth. This included exhibits at many museums (including the National Museum of Computing, Bletchley Park, the Science Museum and the Computer History Museum in Silicon Valley) as well as an issue of a children's computing magazine developed in collaboration with QMUL (Queen Mary University of London) and distributed to UK schools to encourage programming.

Ian's award is for Non-Commercial Impact, and is in recognition of his work with researchers at IIT Kharagpur on the modelling and improvement of filters to remove arsenic from water supplies in India. This work is funded by GCRF (the UK Global Challenge Research Fund) and also supported by UNICEF which is now installing community-scale filters in India. Although it falls outside the definition of the category, Ian is also working with three companies (Dyson, Gore and Pall Corporation) to improve their filters for various purposes.

These awards, which include a £1000 payment, will be presented at the MPLS Winter Reception on February 6th.

There have been reports in the press this week of how the examination length for students taking examinations in the Mathematical Institute at the University of Oxford was extended in summer 2017.

We would like to emphasise that the extension was applied to all students taking those examinations and was for academic reasons. This is part of an ongoing review of our examination processes.

Oxford Mathematician Dan Ciubotaru talks about his recent research in Representation Theory.

"The most basic mathematical structure that describes the symmetries that appear in nature is a group. For example, take all the permutations of the set $\{1,2,3,4\}$ (there are $24$ of them). Each permutation has an inverse one and if you compose two permutations you get another permutation etc. They form the symmetric group $S_4$. On the other hand, you can look at all the rotational symmetries of a cube in $3$-dimensions. There are again $24$ symmetries (this is less obvious) and they interact in the same way as before; in other words, it's the same group $S_4$.

Lie groups, named after the Norwegian mathematician Sophus Lie, are the mathematical objects underlying continuous (in fact, "differentiable'') symmetries inherent in various systems. Examples of Lie groups are the orthogonal groups, e.g., the symmetries of a circle or of a sphere; unlike the $S_4$ example, these are infinite groups. Originally, they appeared in mathematics as groups of symmetries of systems of differential equations, the idea being that by understanding the ways in which these groups act, namely their representations, one can gain some insight into what the space of solutions looks like.

Unitary representations are group actions on Hilbert spaces, at a first approximation, group actions that preserve lengths. The original motivation for their study comes from abstract harmonic analysis (Gelfand's programme) and from quantum mechanics (Wigner's work on the unitary representations of the Lorentz group). Abstract harmonic analysis can be viewed as a vast generalisation of Fourier analysis. In this interpretation, classical Fourier series corresponds to the action of the circle group $S^1$ (the symmetries of the circle) on the space of periodic square integrable functions $L^2(S^1)$. In representation theory, we want to understand the way in which such function spaces decompose into "atoms'' called irreducible representations. This is, as David Vogan from M.I.T. once said, "much as one may study architecture by studying bricks''.

My current research is concerned with understanding unitary representations of reductive $p$-adic groups. A typical example of reductive $p$-adic group is the group of $2$ by $2$ invertible matrices with coefficients in $p$-adic numbers (rather than real numbers). It is harder to explain the occurrence of this group as the symmetries of an object, but it is related to the symmetries of "trees''. To picture what a mathematical tree is, think of the famous sculpture "Y" at Magdalen College (pictured above), except imagine that the tree expands infinitely in all directions. The interest in studying the representations of these groups comes from modern number theory and the theory of automorphic forms. But while this appears to be far removed from 'real life', this theory turned out to have surprising concrete applications: possibly the first interesting construction of a family of expanders, which are certain types of graphs with high connectivity properties very important in computer science, came from this theory.

For these groups, the irreducible representations are almost all infinite dimensional. (Contrast this to the case of $L^2(S^1)$, where the irreducible representations are all one dimensional.) A long-standing conjecture in the field (due to Armand Borel about 1975) essentially asked if one can decide whether or not an infinite-dimensional representation is unitary only by looking at an appropriate finite-dimensional subspace. In a new paper (to appear in Inventiones Mathematicae 2018), I prove this conjecture in general. The new idea is to extend a notion of signature character introduced by Vogan for real reductive groups (and used for $p$-adic groups in an important particular case by Barbasch and Moy) with a "rigid'' density theorem. This latter ingredient stems from joint papers with He (e.g., Journal of the European Mathematical Society 2017), where we are trying to understand the interplay between character theory and a certain natural algebraic object (the cocenter).

For applications to automorphic forms, it is important to have quantitative results about the shape of the "unitary dual'', the set of irreducible unitary representations (together with some natural topological structure). A famous classical result in this direction is Kazhdan's "Property T'' which asserts that the trivial representation (the only one-dimensional representation that all of these groups have) is isolated in the unitary dual. In the second image above, this corresponds to the isolated dots. But for finer applications, one needs to know how isolated is this representation. This is called a "spectral gap.'' It turns out that a very useful tool for determining spectral gaps comes from Dirac operators in representation theory. For real Lie groups, this theory has a distinguished history with many highlights (works of Parthasarathy, Kostant, Atiyah and Schmid, Wallach, Vogan, Huang and Pandzic). For $p$-adic groups, only relatively recently this tool became available when we introduced it in joint work with Barbasch and Trapa (Acta Mathematica 2012) and with Opdam and Trapa (Journal Math. Jussieu 2014). As an application of these ideas, we show in particular that the spectral gap is precisely related to the geometric structure of the nilpotent cone in semisimple Lie algebras. I am currently continuing the research in the applications of the algebraic Dirac operators to representation theory with Marcelo De Martino, a postdoctoral fellow in Oxford, and with my DPhil student Kieran Calvert."

This research is supported by an EPSRC grant. For more on the subject see Dan's webpage.

Oxford Mathematicians Dominic Vella and Finn Box together with colleague Alfonso Castrejón-Pita from Engineering Science in Oxford and Maxime Inizan from MIT have won the annual video competition run by the UK Fluids Network. Here they describe their work and the film.

"We have been studying the wrinkling patterns formed by very thin elastic sheets floating on liquid interfaces to better understand the geometry and mechanics at play. However, to date most interest has focussed on the static properties of these wrinkle patterns: what happens when you gently poke the skin of custard? Here we explore how things change when we are less careful and drop a sphere onto such a film. This shows that the dynamics of this process are different both to normal static wrinkling and to what happens when a stone is dropped into a pond."

We are delighted to announce that Rama Cont has been appointed to the Professorship of Mathematical Finance in the Mathematical Institute here in Oxford. Currently Professor of Mathematics and Chair in Mathematical Finance at Imperial College London, Rama Cont held teaching and research positions at Ecole Polytechnique (France), Columbia University (New York) and Université Pierre & Marie Curie (Paris VI). His research focuses on stochastic analysis, stochastic processes and mathematical modeling in finance, in particular the modeling of extreme market risks.

Professor Cont will take up the post with effect from 1 July 2018.

Oxford Mathematician Ali El Kaafarani explains how mathematics is tackling the issue of post-quantum digital security.

"Quantum computers are on their way to us, not from a galaxy far far away; they are literally right across the road from us in the Physics Department of Oxford University.

All cryptosystems currently used to provide confidentiality/authenticity/privacy rely on the hardness of number-theoretic assumptions, i.e., the integer factorization or discrete logarithm problems. And all was good until a well-known quantum attack, Shor’s algorithm, proved that both these problems can be solved efficiently; this means that once a scalable fault-tolerant quantum computer comes to life, our current cryptosystems become obsolete, and online transactions are deemed completely unsafe.

Cryptographers are currently preparing for that world, which they refer to as the "post-quantum era." Here in Oxford Mathematics, and in collaboration with TU Darmstadt and the University of Tokyo, we are developing post-quantum secure cryptosystems that mainly focus on enhancing and protecting the user's privacy.

At the heart of those cryptosystems is the so-called notion of anonymous digital signatures, which are an authentication method that reveals only the necessary information about the signer, be it a client at a pharmacy who doesn’t want to reveal why he/she doesn’t pay for prescriptions or a machine that wants to prove that it has the right system configuration while communicating with other machines without revealing any further details about itself ; or Internet users who want to stay anonymous while reviewing/rating products.

Those cryptosystems are based on lattices^{1 }which are known to be the most promising quantum-resistant alternative to the number theoretic assumptions in use today. Lattices, those beautiful mathematical objects have come to save the day; they are rich with various sorts of computationally hard problems, namely, worst-case problems (e.g. Shortest Vector Problem), average-case problems (e.g. Short Integer Solution (SIS)^{2}, Learning With Errors (LWE)^{3 }), and most importantly, the relationship between them. Such complexity is what we will need in the future."

An $n$ -dimensional lattice is a (full-rank) discrete additive subgroup of $\mathbb{R}^n$ .

The $\mathsf{SIS}_{n,q,\beta,m}$ problem: given a uniformly random matrix $\textbf{A}\in\mathbb{Z}_q^{n\times m}$ , find a non-zero vector $\textbf{z}\in\mathbb{Z}^m$ of norm bounded by $\beta$ such that $\textbf{A} \textbf{z} = \textbf{0} \in \mathbb{Z}_q^n$ .

At a high level, given a set of "noisy/approximate" linear equations in (a secret) $s\in\mathbb{Z}_q^n$ , the (search) $\mathsf{LWE}$ problem asks to recover $s$.

Global information analytics business Elsevier is donating £1 million to Oxford Mathematics, in support of fellowships, research meetings and workshops.

Oxford Mathematics is widely recognised as one of the foremost centres for the subject globally; its strength and reputation has never been greater. Now, thanks to Elsevier’s generosity, five outstanding early career researchers will be supported by internationally competitive three-year fellowships. Fellows will hold the prestigious title of Hooke or Titchmarsh Fellow; Hooke and Titchmarsh are distinguished figures in the diverse history of Oxford and global mathematics.

During their time at Oxford, fellows will undertake research, develop their experience of teaching in a university environment, and work alongside academics at the forefront of the most profound advances in mathematics. By the end of their fellowships, post-holders will be independent researchers of international standing.

‘We are extremely grateful to Elsevier for this important support for Oxford Mathematics,’ says Professor Martin Bridson, Head of the Mathematical Institute. ‘Postdoctoral fellowships provide vital opportunities to researchers embarking on academic careers. Thanks to this new collaboration, five outstanding early career mathematicians will be supported as they join the institute and pursue some of the most exciting questions in mathematics.’

Finding funding in the current higher education landscape can be extremely challenging for early career researchers. Without support, many individuals struggle to establish an academic career following the end of their doctoral studies. The Mathematical Institute’s Hooke and Titchmarsh Fellowship programme, expanded by virtue of this gift, addresses this need.

Elsevier’s donation will also support a series of high-profile research meetings and workshops at the Mathematical Institute. Spread over the course of five years, the meetings will bring researchers from other UK and international institutions to Oxford in order to engage on topics ranging from data science to fundamental problems in geometry and number theory.

Professor Louise Richardson, Vice-Chancellor of the University of Oxford says: ‘I am delighted that Elsevier has chosen to work with the University’s Mathematical Institute to support our outstanding early career researchers. This initiative will not only benefit researchers here in Oxford but also the international mathematical research community. We are deeply grateful to Elsevier for their generosity.’

Ron Mobed, Elsevier Chief Executive Officer says: ‘The University of Oxford’s commitment to excellence in research, development of young and emerging talent and creating new ways of academic collaboration are very much aligned with Elsevier’s mission. It represents the future of how science should be applied to have a transformative impact on society. Research in mathematics specifically is vital to the exploration of new technologies, innovation, data science and analytics – areas in which we are investing ourselves to make research information more useful.’