# Grant summaries

## Arithmetic of Automorphic Forms and Special L-Values

(EPSRC First Grant, Thanasis Bouganis)

L-functions are known to play a central role in modern number theory. Perhaps the most well known example is the role they play in the famous Birch and Swinnerton-Dyer conjecture. Given an elliptic curve, for simplicity defined over the rational numbers, the conjecture relates various arithmetic invariants of the elliptic curve (such as the rank of the Mordell- Weil group, the size of the conjecturally finite Tate-Shafarevich group) in an astonishingly precise way to the value at s=1 of the L-function attached to the elliptic curve. This last one is defined by putting together local information, over all prime numbers, of the elliptic curve, and as defined makes sense only for the real part of s large enough (actually 3/2) and in order to make sense at s=1 one needs to establish its analytic continuation. This is now known thanks to the celebrated work of Andrew Wiles on the modularity of elliptic curves, and it is achieved by identifying the L-function of the elliptic curve with an L-function of a modular form. Actually the picture just described conjecturally extends to a much general situation. Namely to an arithmetic object, usually called a motive, one associates an L-function which it is believed to encode in its special L-values important information about the underlying motive (Bloch-Kato conjectures). However these L-functions, even though they are defined in the realm of arithmetic geometry, they can be studied with the current status of knowledge only by identifying them with the L- function of an automoprhic form. This connection between motivic and automorphic L-functions suggests that special values of automoprhic L-functions may enjoy interesting arithmetic properties.

Indeed the main aim of this research project is to investigate algebraic and p-adic properties of special L-values of automorphic forms of various kinds, namely Hermitian, Siegel and Siegel-Jacobi modular forms. For Hermitian and Siegel modular forms we aim to the construction of abelian and non-abelian p-adic measures, which constitute an indispensable ingredient in the formulation of the Main Conjectures of Iwasawa Theory (commutative or not), which in turn is the only tool available to tackle the aforementioned relation implied by the Bloch-Kato conjecture between arithmetic invariants and special L-values.

The situation is different with respect to the Siegel-Jacobi forms. Their L-function is not known at present to be identified to an L-function obtained from Galois representations, and actually they are not related to Shimura varieties. However they do enjoy arithmetic structure and in this research grant the aim is to address the question whether the various well-understood phenomena for automorphic forms associated to Shimura varieties (algebraicity of special L-values, Garrett's conjecture on Klingen-type Eisenstein series etc) are still valid for Siegel-Jacobi modular forms.

Research Associate (one year from April 2016): TBA

## Cluster algebras, Coxeter groups and hyperbolic manifolds

(EPSRC Anna Felikson)

Coxeter groups appear in mathematics as symmetry groups of many objects, in particular, symmetry groups of regular polyhedra (the five Platonic solids known already to ancient Greeks) as well as symmetry groups of many tilings used both in art and real life.

Cluster algebras is a very recent notion introduced by Fomin and Zelevinsky in 2002. Soon after that, it turned out that cluster algebras are connected to many other fields in mathematics, such as combinatorics of polytopes, representation theory, Poisson geometry, Teichmuller theory, integrable systems. These connections brought together researchers from many different branches of mathematics and mathematical physics, which induced amazingly rapid growth both of the theory of cluster algebras and of related fields.

It was known since introduction of cluster algebras that some cluster algebras are connected to some Coxeter groups. More precisely, finite cluster algebras (the only ones which are finitely generated) are enumerated by finite crystallographic Coxeter groups.

The first aim of this proposal is to push this correspondence further to the next complexity class of cluster algebras (called cluster algebras of finite mutation type and including a large class of cluster algebras arising from triangulated boardered surfaces). Algebras from this class should correspond to certain quotients of Coxeter group. For algebras arising from surfaces, the relations in the constructed group should correspond to certain paths on the surface.

There are many natural questions arising once the correspondence between algebras and groups constructed. In particular, it is interesting to know if some of the groups constructed as quotients of Coxeter groups are Coxeter groups themselves? Can the constructed group be finite if the cluster algebra is not a finite one? Do different algebras induce different groups? How the obtained group is connected to the initial surface, in the case of a surface algebra?

As one of the applications of the theory, we will construct finite volume hyperbolic manifolds with large symmetry groups.

Research Associate: Ilke Canakci (2 years from October 2015).

## Particles, Fields and Spacetime

(STFC Simon Ross)

Particle Physics has entered a new and crucial phase. The Large Hadron Collider at CERN has enabled us to examine experimentally theoretical concepts that underly the standard model of particle physics, such as the Higgs mechanism, and it will go on to search for the deeper structures that are believed to unify the laws of physics.

Quantum field theory is the mathematical language in which the standard model is expressed, and it treats particles as point-like objects. Only certain kinds of quantum field theories, known as gauge theories, are consistent in the four dimensional world we live in. These include, and are generalisations of, the theory of electrodynamics that describes light interacting with electric charge. To be able to interpret the results of experiments we need to be able to solve gauge theories, at least approximately. This is a hard problem, but one in which there has recently been very remarkable progress due to a convergence of ideas originally developed in quite disparate contexts for solving very different kinds of theories. A major thrust of the project will be to push this line of enquiry further so as to be able to more fully understand gauge theories and be able to compute their properties.

Matter at large scales is dominated by gravity which is described by Einstein's theory of General Relativity. This governs the motion of planets, stars, galaxies, and the evolution of the Universe itself. Uniting General Relativity and the standard model of particle physics is the most important challenge facing theoretical physics. It is widely, though not universally, believed that string theory provides such a unification. String theory replaces the point-like particles of quantum field theory with extended objects whose different vibrational modes account for the different species of fundamental particles. It is this belief that leads to the expectation that supersymmetry, a property of all realistic string theories, plays a role in nature, and may well be discovered at the LHC. Showing how nature contrives to hide this property is another part of the project. String theory has also led to many unexpected relations between different kinds of physical theories, most notably in the AdS/CFT correspondence which states equivalences between certain gravity theories and corresponding gauge theories, enabling us to solve difficult problems in one theory by studying simper ones in the other. We will use this to study problems in gravity that would otherwise be intractable and also model strongly coupled physical processes in diverse areas ranging from condensed matter physics to plasmas by gravity.

We will also use another method for studying hadrons that is particularly appropriate to describing large numbers of thembound into nuclei or even neutron stars. This is based on effective field theories such as the Skyrme model which we will investigate numerically using computers.

Being a theory of quantum gravity strings have many implications for cosmology, in particular they admit the possibility that what we see as the physical universe is only a low dimensional subspace called a brane, moving in a space of higher dimensions. We will continue the quest to find direct experimental and observational signatures that will test this scenario.

The grant is awarded to the members of the Centre for Particle Theory who are also staff in the Department of Mathematical Sciences.

## New homotopy-type invariants of knots

(EPSRC, Andrew Lobb)

To a mathematician knots are essentially the same as they are to anyone - tangled up pieces of string. Studying when two knots are different turns out to have deep implications for a wide range of mathematical areas. This project seeks to develop a new way to distinguish between knots and will have consequences for our understanding of three and four dimensional objects.

Research Associate: Patrick Orson (from January 2015).

## Photospheric Driving of Non-Potential Coronal Magnetic Field Simulations

(US Air Force Grant, Anthony Yeates)

Forecasting the arrival at Earth of interplanetary coronal mass ejections from the Sun relies on models both for their propagation and for the background solar wind. In particular, both the ejections and the background solar wind are driven by processes in the inner solar corona. This project aims to develop an alternative to the potential field extrapolation that is currently used for the innermost model component in operational space-weather forecasts. Due to its assumption of zero electric current density, the potential field model precludes the formation of twisted flux rope structures, so cannot predict the magnetic topology of ejecta. There is also evidence that neglecting these currents may lead to innaccuracies in the ambient solar wind model.

The idea of the project is to develop a new photospheric driver to assimilate high-resolution radial magnetic maps into non-potential "magneto-frictional" simulations of the coronal magnetic field. This will be applied with input data from the Air Force Data Assimilative Photospheric Flux Transport Model. We aim to assess how the magneto-frictional model can improve background solar wind forecasts and whether flux ropes in the model can constrain the magnetic topology of ejections.

Research Associates: Sarah Edwards and Marion Weinzierl.

## SPOCK: Scientific Properties Of Complex Knots

(Leverhulme Trust Research Programme Grant, Paul Sutcliffe)

The complexity of three-dimensional systems is often topological: spatial systems of filaments, dynamically evolving, typically become tangled and knotted in a way that reveals underlying properties of the system. The aim of the project is to create new computational tools and mathematical techniques for the analysis, synthesis and exploitation of knotted structures in a wide range of complex physical phenomena, allowing the development of a deep understanding of topological complexity in nature. This will require new mathematical techniques beyond traditional (abstract) knot theory, which will be developed as a generic Topological Toolkit to synthesize and analyse knotted physical systems. The main question is where and how knots arise in nature, including how the background environment (designed or otherwise) can influence the generation of knotted structures. A driving force for this development is new analytical, numerical and experimental techniques in a number of carefully selected interdisciplinary sub-projects, where experimentalists, theorists, mathematicians and scholars from the humanities can work together in identifying and solving problems in applied knot theory.

This collaboration between Durham University and the University of Bristol, with the award including funding for 4 Research Associates for 4 years and 8 PhD students, divided equally between the two Universities.

- Durham Research Associates: Kate Horner and Fabian Maucher.
- Durham Students: David Evans.

## SPiN: Symmetry Principles in Nature

(ERC Consolidator Grant, Mukund Rangamani)

Symmetries have traditionally played a very important role in our understanding of physics, both classical and quantum. As we move towards the next frontier of defining a quantum theory of gravity, it is clear that they will continue to play a predominant role. The current project is aimed at obtaining a comprehensive understanding of the dynamics of strongly coupled quantum systems, exploiting various symmetry properties that one expects on physical grounds. Specifically, the aim is to come up with effective descriptions of a wide class of quantum dynamics in gravitational and non-gravitational theories using the holographic gauge/gravity correspondence.

One of the primary strands of the proposed research involves a critical examination of gravitational theories with higher spin symmetry. We will investigate the phase structure of such theories, the nature of gravitational solutions and notions of classical geometry in the presence of the enlarged gauge symmetry. Using appropriate generalizations of the gauge/gravity correspondence we will try to give gauge invariant characterizations of these concepts and further explore how such higher spin theories can be realized in string theory. A related strand of research concerns a deeper understanding of the gauge/gravity correspondence itself, with focus on figuring out how collective behaviour of strongly coupled field theories leads to the emergence of extra symmetries, such as local diffeomorphisms in the dual description. Along the way we will also develop effective descriptions for collective dynamics of strongly coupled quantum degrees of freedom, both in and out of equilibrium.

Research Associate: Wei Li

## Measures of Holographic Information

(Foundational Questions Institute, Veronika Hubeny)

Whilst information is a concept we have natural intuition for in everyday circumstances, it is less tangible when one asks questions in a microscopic quantum theory. In recent years the subject of quantum information has grown significantly and now one has the mathematical tools to quantify information in a quantum mechanical system. At the same time, inspired by developments in string theory and black holes, one has learnt that gravitational theories are holographic. This means that one can describe them by completely different quantum mechanical degrees of freedom. This is at the heart of the famous gauge/gravity correspondence which treats the gravitational universe as a hologram. While the correspondence has been immensely useful in characterizing many aspects of gravitational physics and also quantum dynamics, we are far from having a complete understanding of this holographic picture. The project aims to employ techniques from the realm of quantum information to address our ignorance about the holographic map. In an explicit sense we would like to crack this holographic code and by doing so unearth the fundamental aspects of how spacetime, geometry, and quantum dynamics are unified in a complete theory.

Co-Investigator: Mukund Rangamani

## Theory of Badly Approximable Sets

(EPSRC First Grant, Dzmitry Badziahin)

A given real number can be approximated by rational numbers with an arbitrary precision. For example the decimal representation of a number provides such an approximation. On the other hand different numbers are approximated by rationals with different speed in terms of the denominators of that rational numbers. There is no upper bound for that speed which was firstly mentioned by Liouville in 1840's. On the other hand there is a natural lower bound for that speed which is determined by Dirichlet-Hurwitz theorems. It says that for any irrational number x one can find infinitely many rational numbers p/q such that |x - p/q| < 1/q^2. Moreover there exist irrational numbers which make this inequality sharp. In other words for that numbers the distance |x-p/q| can not be made smaller than c/q^2 for some positive constant c and all rational numbers p/q. Irrational real numbers with such property are called badly approximable. The set of badly approximable numbers can be described quite well with help of the theory of continued fractions. It allows us to derive a lot of information about that set: it's "size" in terms of Lebesgue's measure and Hausdorff dimension and some other information about its structure. In two-dimensional case and more generally in high dimensional case we can also approximate any point in R^n by points with rational coefficients. However different coordinates can be approximated by rationals with their own speed. Because of this phenomena, in higher dimensions we have an uncountable family of different sets of badly approximable points depending on the relative speed of approximation of different coordinates of the point. The structure of the sets of badly approximable points in high dimensions is much less known than their one-dimensional analogue. It incorporates a lot of open problems. Some of them like famous Littlewood conjecture attract a lot of attention of the modern mathematical society. In this project I am going to shed the light on many of that problems.

One of the main objectives of this project is to construct the mechanism which, like continued fractions in one dimension, will help us to deal with badly approximable points easier. This work has already been started by me together with Velani and Pollington. We described the sets which allowed us to solve a number of related problems including the famous Schmidt conjecture. Next, the mechanism of generalized Cantor sets will be used to solve some particular open problems about badly approximable points in n-dimensional real space. In particular the structure of BAD points on manifolds in arbitrary dimensions will be investigated. Finally some new approaches will be applied to attack the famous Littlewood conjecture. It was stated by Littlewood in 1931 and nowadays it is one of the most attractive problems in modern Mathematics.

Research Associate: Stephen Harrap.

## Topology, Geometry and Laplacians of Simplicial Complexes

(EPSRC, Norbert Peyerimhoff)

Simplicial complexes are natural abstract mathematical objects which play a prominent role in several fields of mathematics. They appear as triangulations of surfaces or more general higher dimensional spaces, which are useful in topology for the computation of invariants like the Euler characteristic. Other important examples of simplicial complexes, in connection with geometric group theory, are buildings. They were first introduced by Jacques Tits in his work on Klein's Erlangen program and provide a very successful geometric approach to group theory.

Being combinatorial objects, simplicial complexes can serve as simplified models of smooth geometric spaces. Their combinatorial nature allows explicit computations of their fundamental groups. Fundamental groups are a basic algebraic tool to describe the equivalence of closed paths under continuous deformations. In this project, we aim to obtain a better understanding of the fundamental groups of simplicial complexes and their properties. Another attractive property of simplicial complexes is that they can be endowed with additional geometric structures and become gateways to a much richer world than the classical surfaces and their generalisation to higher dimensions (manifolds). In this project, we aim is to generalize known geometric concepts of curvature into this richer world. Another consequence of the simplicity and versatility of simplicial complexes is their wide use as geometric representations in the fields of industrial design and medical imaging. The better understanding of their mathematical properties will lead to improved processing algorithms that can be used in these applications.

3-year postdoc: Dr Shiping Liu

## Complex Magnetic Fields: An Enigma of Solar Plasmas

(STFC, Anthony Yeates)

The outer atmosphere of the Sun, the solar corona, is a dynamic plasma permeated by a magnetic field. This magnetic field is responsible for creating long-lived structures such as coronal loops, for heating the corona to its multi-million degree temperatures, and for explosive events such as solar flares and coronal mass ejections. Understanding how these explosive processes take place requires a detailed understanding of the behaviour of the Sun's magnetic field, which is characterised by its complex three-dimensional structure. Gaining such an understanding is the aim of this work programme and is part of a wider goal in the scientific community of understanding the formation of structures in astrophysical plasmas.

Recent observational advances are providing us with a more and more detailed view of the Sun's magnetic field. But each increase in spatial resolution reveals finer scale magnetic structures, down to the resolution limit of even the most advanced telescopes. What is more, each increase in time cadence reveals more complex dynamics that shape the magnetic field and plasma on all scales. The over-arching theme of this consortium proposal between the universities of Dundee and Durham is to explore the physical consequences of this magnetic complexity. We aim to understand how such complex magnetic fields are formed, how they evolve, and how they can build up and explosively release extreme amounts of energy. We will use a combination of numerical simulations and mathematical modelling to tackle these questions, primarily using the non-linear partial differential equations of magnetohydrodynamics. The modelling will take input from the latest generation of solar telescopes, using various observations to verify and refine the theory.

Durham is the lead institution on a sub-project to address two questions. Can we develop new tools to help analyse these complex magnetic fields, and can we apply these tools to the evolution of the corona on global scales?

Research Associate: Sarah Edwards/ Christopher Lowder.

## Representation Growth of Linear Groups over Local Rings

(EPSRC, Alexander Stasinski)

Zeta functions play a major role in number theory, starting with the Riemann zeta function, and continuing with Dedekind zeta functions and more general Dirichlet series. More recently zeta functions have started to play a role also in group theory and representation theory. In these cases the zeta function encodes an infinite sequence of numbers a_{1}, a_{2},..., where a_{n} is, for instance, the number of irreducible n-dimensional representations of a group. The zeta function defined using this sequence is called the representation zeta function of the group. Among other things, the zeta functions help us to say something about the growth rate of the sequence a_{1}, a_{2},.... For large classes of interesting groups the precise rate of growth is governed by a rational number called the abscissa of convergence of the zeta function.

The first part of this project will study the abscissa of convergence associated to the groups SL_{N}(o), that is, the group of N by N matrices with determinant 1 and with entries in a local principal ideal ring with finite residue field, such as the p-adic integers Z_{p}. The precise value of the abscissa of SL_{N}(o) is a mysterious object and there is currently not even a guess as to its precise value. We will develop and use several constructions for representations of the related group GL__{N}(o) of invertible matrices over o to pin down the value of the abscissa more precisely than has been done previously. A prominent role here is played by the so-called regular representations.

The first goal is to find a precise conjectural value of the abscissa and to prove it for as many groups as possible. Since our methods work for all rings o, the second goal will be to address the problem of whether or not the abscissa of SL_{N}(o) is independent of o, where N and the residue field of o is fixed but o varies.

The second part of the project will generalise certain results from classical Deligne-Lusztig theory to the generalised unramified Deligne-Lusztig construction. These results will then be used to prove a generalisation of an asymptotic formula of Liebeck and Shalev for Chevalley groups over finite local rings G(o_{r}). There is some evidence that this formula can be viewed as a finite group analogue of some formulae for the abscissa of G(o), but the precise explanation for this is still not known, and our work aims to shed some light on this.

Finally, we aim to tie together the above topics by showing that the representations given by the generalised unramified Deligne-Lusztig construction for GL_{n}(o_{r}) are regular, and have therefore been included in the counting in the first part of the project.

Research Associate: Jokke Häsä.

## Knot Homology: Theory and Computation

(EPSRC, Andrew Lobb)

Knot homologies are a hot topic in low-dimensional topology. Much of the activity, however, is focused on construction and less on understanding. This grant has enabled Durham to employ a research associate to work with Andrew Lobb in a project to understand a large class of these knot homologies, both through theoretical arguments and by direct computation.

Research Associate: Lukas Lewark

## Non-homogeneous Random Walks

(EPSRC, Andrew Wade)

Random walks are fundamental models in stochastic process theory that exhibit deep connections to important areas of pure and applied mathematics and enjoy broad applications across the sciences and beyond. Specific application areas include for example modelling of microbe locomotion in microbiology, polymer conformation in molecular chemistry, and financial systems in economics.

Spatially homogeneous random walks, in which the probabilistic nature of the jumps is the same regardless of the present spatial location of the walker, are the subject of a substantial literature. In many modelling applications, the classical assumption of spatial homogeneity is unrealistic: the behaviour of the random walker may depend on the present location in space. Applications thus motivate the study of non-homogeneous random walks. Moreover, non-homogeneous random walks are the natural setting in which to probe near-critical behaviour and obtain a finer understanding of phase transitions present in the classical random walk models.

The proposed research is part of a broad research programme to analyse near critical stochastic systems. Non-homogeneous random walks can typically not be studied by the techniques generally used for homogeneous random walks: new methods (and, just as importantly, new intuitions) are required. Naturally, the analysis of near-critical systems is more challenging and delicate than that for systems that are far from criticality. The methodology is based on martingale ideas. The methods are robust and powerful, and it is to be expected that methods developed during the project will be applicable to many other near-critical models, including those with applications across modern probability theory and beyond, to areas such as queueing theory, interacting particle systems, and random media.

Research Associate: Nicholas Georgiou

## Geometry, Holography and Skyrmions

(EPSRC, Paul Sutcliffe)

Particle physicists have a good understanding of the fundamental constituents of matter, but the mathematical complexity of their theory means that, even with the use of supercomputers, it is impossible to use it to predict even the basic properties of the simplest atoms familiar from everyday life, such as helium, carbon and oxygen. Fathoming the core of these atoms is the realm of nuclear physics, but current approaches are detached from fundamental theory and instead are mainly based on fitting phenomenological models to experimental data. The ambitious aim of this project is to provide a link between fundamental theory and nuclear physics through a novel mathematical description, using a geometrical formulation, in which atomic nuclei arise as stable localized excitations called Skyrmions.

Research Associates:

- Stefano Bolognesi
- Petja Salmi
- Mike Gillard

## Elliptic Genera and Mathieu Moonshine

(The Leverhulme Trust, Anne Taormina)

The Mathieu Moonshine observation refers to an unexplained relation between the largest Mathieu simple sporadic group M24 and the elliptic genus of K3 surfaces, which may be viewed as a specialization of the partition function of superstrings compactified on K3 surfaces. More specifically, the Fourier coefficients of the elliptic genus have an interpretation as the dimensions of representations of M24.

The first implication of this observation is that the group M24 acts on the string states selected by the K3 elliptic genus, which is constant on the moduli space of N=(4,4) superconformal field theories at central charge c=6, whose symmetry groups are much smaller than M24. The current challenge is to understand the mechanisms of this M24 action, which manifests itself in a highly unconventional manner in string theory. Although the Mathieu Moonshine shares similarities with the Monster Moonshine, its explanation appears to require a different blend of techniques from number theory (mock modular forms), algebraic geometry, group theory and conformal field theory, and the explanation of the Mathieu Moonshine phenomenon is likely to open interesting new perspectives, not only on string theories compactified on K3 surfaces, but also on other string theories, and in the mathematical disciplines listed above.

The purpose of this 12-month grant is to make progress in the understanding of the Mathieu Moonshine phenomenon by exploiting its connection with string theory. Some recent progress is summarized here.

## Analysis & Computation of Microstructure in Finite Plasticity

(Deutsche Forschungsgemeinschaft, Patrick Dondl)

Plastic behaviour is ultimately related to the motion (and its inhibition) of individual dislocations. While there has been considerable progress in the computational modeling of the motion of individual dislocations and the local interaction of different dislocations, the large number of dislocations in a given specimen requires a multiscale approach to understand the mesoscopic and macroscopic effects of dislocations.

In particular, dislocations in plastically deformed crystals exhibit intricate sub-grain structures which in turn greatly affect the plastic properties of the medium. In this project we want to answer fundamental questions on how these structures arise and how they affect the macroscopic behavior of the crystal. The main goal is to mathematically rigorously derive predictions from certain classes of models that result in microstructure and compare those predictions with experiments in an interdisciplinary collaboration. This comparison will allow us to infer refined macroscopic models for plastic deformation in order to account for the relevant microstructures.

Research Associate: Keith Anguige

## Persistent topological structures in noisy images

(EPSRC, Vitaliy Kurlin)

The proposed research is on the interface between topology and algorithms.The methods are topological coming from the recently developed theory of persistent topology.The key idea of persistent topology is identifying features of geometric objects that remain stable for a wide range of varying parameters. The research objectives are applied and involve designing novel learning algorithms to recognise topological graphs in noisy images.A natural application is identifying a text-based advanced CAPTCHA (Completely Automated Public Turing test to tell Computers and Humans Apart) widely used on the web to prevent automatic posting to blogs.A desired learning algorithm will be parallel like the human brain analysing local persistent properties of pixels in a noisy image.Hence it is the next step towards reverse-engineering the human brain.Other potential applications include recognising handwritten mail addresses and designing computer glasses reading signs of directions for blind people.

1-year postdoc: Alexey Chernov

## Integrability, Symmetry and Quantum Space

(EU, Douglas Smith)

String theory is the prime candidate for a consistent theory of quantum gravity. It should therefore be able to address fundamental questions such as the origin and structure of spacetime. The key to an understanding of these deep problems is to uncover the symmetry principles that underlie string and to gain control of string theory in its non-perturbative regime. The primary scientific goal of this project, “Integrability, Symmetry And Quantum Spacetime” (ISAQS) is to uncover new techniques in the theory of integrability that will be useful for the understanding of integrability in string theory, as well as to discover and study new symmetry principles of string theory, so as to obtain better control and new insights about the quantum structure of spacetime at high energies or small scales. Our scientific objectives come under 3 workpackages:

• Integrability of String Theory – To understand the appearance and role of integrability in string theory. This can give new techniques to calculate the spectrum of states in string theories, and through gauge/gravity duality such developments would allow us to determine the full spectrum of field theories such as N = 4 Super-Yang-Mills (SYM), as well as giving information about scattering amplitudes and their relation to Wilson loops.

• Techniques of Integrability – With renewed interest in, and new examples of, integrability in string theory contexts, it should be very productive to generalise techniques such as the thermodynamic Bethe-Ansatz and nonlinear integral equations so that these powerful methods can be applied in more general contexts, such as four-dimensional SYM theory. We are also searching for new examples of integrable systems by exploring the classes of possible boundary conditions and defects which can be included in known models.

• Membranes and New Symmetry of String Theory – While D-branes in string theory are well understood and have been used in many applications due to their relation to SYM theories, the theory of branes in M-theory is less well developed. Progress in describing multiple M2-branes has uncovered novel symmetries, and we are working to develop our understanding of these branes and the even less well understood M5-branes. The results are expected to give insight into the nature of M-theory and deepen our understanding of field theories in various dimensions. This project provides support for researchers from the two EU partner organisations at Durham University, UK and the Albert Einstein Institute, Potsdam, Germany, to visit the two third country partner organisations at CQUeST, Seoul, Korea and YITP, Kyoto, Japan. Researchers at CQUeST have reciprocal funding. These exchanges have enabled transfer of knowledge and collaborations to be initiated and developed.

## A Bayesian Approach to Some Deconvolution Problems for oil and gas reservoirs

(Industry Consortium, David Wooff)

With current trends towards intelligent wells and fields, continuous bottomhole well pressure monitoring is becoming the norm in new field developments. Theoretically, this should allow operators to better control well performance and address problems before they become irreversible. In practice, the need to interpret the raw information provided by permanent gauges, and the lack of manpower or expertise for doing so prevents real time intervention. Therefore, some sort of automatic interpretation and alarm system is required to benefit from the full potential of downhole permanent gauges. Deconvolution of flow and pressure histories, based on an algorithm developed at Imperial College, provides a viable possibility.

The purpose of this research is to introduce a proper stochastic element to the single-well problem, and then to extend a stochastic version to solve the multi-well problem. To do this, we use Bayesian statistics in order to obtain a robust multiwell deconvolution algorithm that can be used by practicing engineers in a wide variety of field conditions to control and optimise well performance. A second objective is to investigate the use of Bayesian statistics for well test interpretation model identification, in order to obtain estimates of reservoir properties and future flow/pressure behaviours with stated levels of accuracy.

This research is collaborative with Imperial College, London, via the Joint Industry Project on Continuous Interpretation of Well Test Data by Deconvolution, funded by Anadarko, BG, Chevron, ConocoPhillips, Inpex, Petro SA, Saudi Aramco, Schlumberger, Shell, and Total.

Research Associate: Jonathan Cumming

## Hidden Beauty of Gauge Theories: Geometry & Integrability

(EPSRC, Marija Zamaklar)

Most of modern quantum field theory is based on the remarkable framework of Yang and Mills, who used structures that also occur in geometry to describe the dynamics of elementary particles. However, despite the fact that the predictions of this theory have been rigorously tested experimentally, its mathematical foundation is still not fully understood. Namely, an important open problem exists on how to analytically compute the spectrum of this theory at strong coupling. Though there are computer simulations which reproduce the observed spectrum, a theoretical understanding of it is still missing. The main obstacle in computing the spectrum of Yang-Mills theory, is that one needs to understand a system at strong coupling; no natural small parameter exists which would allow for a standard perturbative approach. Some ten years ago, an ingenious conjecture was proposed by Maldacena on how to address this long-standing problem. The idea is very simple, though conceptually challenging: instead of analysing the theory in four space-time dimensions of our world, he proposed to consider string theory in a higher-dimensional, curved space. He conjectured that this higher-dimensional string theory is equivalent to the lower-dimensional theory of Yang and Mills and proposed a specific map between the two. The key point of this map is that it inverts the coupling between the two theories: hence strongly coupled (and hard to address) phenomena in one theory are mapped to weakly coupled (and easy to compute) phenomena in the other theory.

Though the conjecture has so far been checked in specific limits, its proof is still lacking. Recent discoveries of integrable structures in maximally supersymmetric Yang-Mills theory (N=4 SYM) have introduced new ideas on how one could prove the conjecture. The main goal of my research is to work towards a proof of the string/gauge theory correspondence using these insights. The methods which will be used require a combination of several interdisciplinary techniques coming from integrability, quantum field theory, string theory and group theory. As such, the project is very challenging and a construction of the proof is likely to lead to new developments in mathematics and physics.

Although the initial conjecture was formulated for a very specific case of Yang-Mills theory (the N=4 SYM theory) there are strong indications that the ideas should hold much more generally. Pushing the limits of the conjecture and understanding where it breaks is another goal of my research. Since the conjecture opened up a conceptually new way of looking at field theory and gravitational (i.e. geometrical) phenomena, understanding to what kind of systems it can be applied will teach us about fundamental properties of Yang-Mills theories.