Wednesday, 24 January 2018

Oxford Mathematics Research studies Unitary representations of Lie groups

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.

Tuesday, 23 January 2018

Wrinkly Impact - award-winning film from Oxford Mathematics

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."


Wednesday, 17 January 2018

Rama Cont appointed to the Professorship of Mathematical Finance in Oxford

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.

Tuesday, 16 January 2018

New Cryptosystems to ensure Data Security in the Post-Quantum World

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 latticeswhich 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)), and most importantly, the relationship between them.  Such complexity is what we will need in the future."

The Public Key Cryptography 2018 and The Financial Cryptography and Data Security 2018 Conferences will feature some of this work.

  1. An $n$ -dimensional lattice is a (full-rank) discrete additive subgroup of $\mathbb{R}^n$ .
  2. 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  $\textbf{A} \textbf{z} = \textbf{0} \in \mathbb{Z}_q^n$ .
  3. 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 $s$.
Tuesday, 16 January 2018

Elsevier supports mathematics at Oxford with £1 million donation

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.’

Support Mathematics at Oxford.

Monday, 15 January 2018

Gauge theory - where particle physics and mathematics meet


Oxford Mathematician Yuuji Tanaka describes his part in the advances in our understanding of gauge theory.

"Gauge theory originated in physics, emerging as a unified theory of weak interaction such as appears in beta decay and electro-magnetism via the framework of Yang-Mills gauge theory together with the "Higgs mechanism" which wonderfully attaches mass to matters and forces. It became a mainstream of particle physics after the great discovery of the renormalisable property by Veltman and 't Hooft, and it gives precise descriptions of the experiments. Nowadays all fundamental interactions (electro-magnetism, weak force, strong force, gravity) can be described by gauge theory.

These development certainly stimulated the mathematical studies of gauge fields, particularly in the field of principal or vector bundles. In this context, the curvature of a connection corresponds to the field strength of a gauge field. In the early 80s Donaldson looked into the moduli space of solutions to a certain type of Yang-Mills gauge field (called self-dual or anti-self-dual connections), and produced surprising ways to distinguish differential structures of 4-dimensional curved spaces with the same topology type by using the moduli space or by defining invariants of smooth structures through the moduli space.

After Donaldson's work, Witten skilfully reinterpreted it in terms of a certain quantum field theory. Subsequently Atiyah and Jeffrey mathematically reformulated Witten's work by using the Mathai-Quillen formalism. These exchanges of ideas were one of the stepping stones which led to the discovery of the Seiberg-Witten equations and their invariants around 1994 through a generalisation of the electro-magnetic duality, a hidden symmetry in the theory of electro-magnetism. Seiberg and Witten presented a striking application of this on a super Yang-Mills theory in the quantum level, called strong-weak duality, which enables one to calculate things in the strong coupling region of the theory in terms of ones in the weak coupling region.

Vafa and Witten further analysed Seiberg and Witten's work in a more symmetric model, and conjectured that the partition function of the invariants in this case would have a modular property, which is a mathematical enhancement of the strong-weak duality mentioned above and originally discovered in the theory of elliptic curves in the 19th century. They examined this property in examples by using results from mathematics under the assumption that the Higgs fields automatically vanish.

However, even a mathematically rigorous definition of this, especially including Higgs fields, was not produced for over 20 years. Richard Thomas and myself recently defined deformation invariants of projective surfaces, from the moduli space of solutions to the gauge-theoretic equations of the Vafa and Witten theory, by using modern techniques in algebraic geometry. We then computed partition functions of the invariants coming from non-vanishing Higgs fields as well. Surprisingly, our calculations match the conjecture by Vafa and Witten more than two decades ago despite ours also including sheaves on the surface."

Click here and here for the published research.

Friday, 12 January 2018

Abel in Oxford - Lectures by Abel Prize winners and members of the Abel Prize Committee, Oxford, 15 January

The Abel Prize is the most prestigious prize in Mathematics. Each year, in anticipation of the prize announcement, an afternnon of lectues showcases previous winners and member of the Committee. This year the event will be held in Oxford on Monday 15th January. Andrew Wiles, John Rognes and Irene Fonseca will be the speakers. Full details below. Everyone welcome.  No need to register.


1.00pm: Introductory Remarks by Camilla Serck-Hanssen, the Vice President of the Norwegian Academy of Science and Letters

1.10pm - 2.10pm: Andrew Wiles

2.10pm - 2.30pm: Break

2.30pm - 3.30pm: Irene Fonseca

3.30pm - 4.00pm: Tea and Coffee

4.00pm - 5.00pm: John Rognes


Andrew Wiles: Points on elliptic curves, problems and progress

This will be a survey of the problems concerned with counting points on elliptic curves.


Irene Fonseca: Mathematical Analysis of Novel Advanced Materials

Quantum dots are man-made nanocrystals of semiconducting materials. Their formation and assembly patterns play a central role in nanotechnology, and in particular in the optoelectronic properties of semiconductors. Changing the dots' size and shape gives rise to many applications that permeate our daily lives, such as the new Samsung QLED TV monitor that uses quantum dots to turn "light into perfect color"! 

Quantum dots are obtained via the deposition of a crystalline overlayer (epitaxial film) on a crystalline substrate. When the thickness of the film reaches a critical value, the profile of the film becomes corrugated and islands (quantum dots) form. As the creation of quantum dots evolves with time, materials defects appear. Their modeling is of great interest in materials science since material properties, including rigidity and conductivity, can be strongly influenced by the presence of defects such as dislocations. 

In this talk we will use methods from the calculus of variations and partial differential equations to model and mathematically analyze the onset of quantum dots, the regularity and evolution of their shapes, and the nucleation and motion of dislocations.


John Rognes: Symmetries of Manifolds

To describe the possible rotations of a ball of ice, three real numbers suffice.  If the ice melts, infinitely many numbers are needed to describe the possible motions of the resulting ball of water.  We discuss the shape of the resulting spaces of continuous, piecewise-linear or differentiable symmetries of spheres, balls and higher-dimensional manifolds.  In the high-dimensional cases the answer turns out to involve surgery theory and algebraic K-theory.

Friday, 12 January 2018

Oxford Mathematician Sir John Ball awarded the King Faisal Prize for Science

Oxford Mathematician Sir John Ball FRS has been awarded the King Faisal Prize for Science. Launched by the King Faisal Foundation (KFF) and granted for the first time in 1979, the King Faisal Prize recognises the outstanding works of individuals and institutions in five major categories: Service to Islam, Islamic Studies, Arabic Language and Literature, Medicine, and Science. Its aim is to benefit Muslims in their present and future, inspire them to participate in all aspects of civilisation, as well as enrich human knowledge and develop mankind.

Sir John Ball is Sedleian Professor of Natural Philosophy, Director of the Oxford Centre for Nonlinear Partial Differential Equations and Fellow of the Queen's College. John's main research areas lie in the calculus of variations, nonlinear partial differential equations, infinite-dimensional dynamical systems and their applications to nonlinear mechanics.

Friday, 12 January 2018

The mathematics of security - Oxford Mathematics researchers and undergraduates expose security flaw

As recent breaches have demonstrated, security will be one of the major concerns of our digital futures. The collective intelligence of the mathematical community is critical to finding these flaws. A group of Oxford Mathematicians, both researchers and undergraduates, have done just that.

SecureRF is a corporation founded in 2004 specialising in security for Internet of Things (IoT), i.e. devices with low processing power that require ultra-low energy consumption, whose partners include the US Air Force. SecureRF has also collaborated with Intel to develop an implementation of WalnutDSA on secure field-programmable gate arrays. WalnutDSA (trademarked by SecureRF) is an example of  a digital signature algorithm, a mathematical scheme for demonstrating the authenticity of digital messages (like a real signature, but digital). Walnut DSA uses high-level mathematical techniques from permutation groups, matrix groups and braid groups, and is designed to provide post-quantum security in lightweight IoT device contexts.

The Oxford Team attacked the algorithm by bypassing the E-Multiplication and cloaked conjugacy search problems at its heart, forging signatures for arbitrary messages in approximately two minutes. Thanks to this cryptanalysis, the scheme has now been modified accordingly and an upgrade that corrects the security risk submitted to the National Insititute of Standards and Technology (NIST) competition for Post Quantum Cryptography.

One of the most pertinent and inspiring parts of the story is that the exposure was the result of a collaboration between researchers Giacomo Micheli and Christophe Petit and undergraduates Daniel Hart, DoHoon Kim, Guillermo Pascual Perez and Yuxuan Quek - the work was one of the summer projects that Oxford Mathematics uses to develop and inspire its undergraduate mathematicians, giving them a taste of rigorous research. 

A fuller explanation can be found here and will be presented at PKC 2018, the 21st edition of the International Conference on Practice and Theory of Public Key Cryptography.

Thursday, 11 January 2018

Oxford Mathematician Sarah Waters awarded a Leverhulme Trust Senior Research Fellowship

Oxford Mathematician Sarah Waters has been awarded a Royal Society Leverhulme Trust Senior Research Fellowship commencing this month. Sarah is an applied mathematician here in Oxford. Her interest is in physiological fluid mechanics, tissue biomechanics and the application of mathematics to problems in medicine and biology. Her work varies from classical applied mathematics problems motivated by physiological applications to highly interdisciplinary work - she collaborates with life scientists, clinicians, bioengineers, theoreticians and experimentalists to develop and solve models that are novel, realistic and provide insights into biomedical problems. The resulting models often lead to theoretical predictions that can be exploited in the laboratory.