You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.
Title  Classical and quantum Chaos in 3body Coulomb problems 

Group(s)  Industrial and Applied Mathematics, Mathematical Physics 
Proposer(s)  Prof Gregor Tanner 
Description  The realisation that the dynamics of 2 particles interacting via central forces is fundamentally different from the dynamics of three particles can be seen as the birth of modern dynamical system theory. The motion of two particles (for example the earthmoon problem neglecting the sun and other planets) is regular and thus easy to predict. This is not the case for three or more particles (especially if the forces between all these particles are of comparable size) and the resulting dynamics is in general chaotic, a fact first spelt out be Poincaré at the end of the 19th century. An important source for chaos in the threebody problem is the possibility of triple collisions, that is, events where all three particles collide simultaneously. Triple collisions form essential singularities in the equation of motions, that is, trajectories can not be smoothly continued through triple collision events. This is related to the fact, that the dynamics at the triple collision point itself takes place on a collision manifold of nontrivial topology. During the project, the student will be introduced to scaling techniques which allow to study the dynamics at the triple collision point. We will in particular consider threebody Coulomb problems, such as twoelectron atoms, and study the influence of the triplecollision on the total dynamics of the problem. As a long term goal, we will try to uncover the origin of approximate invariants of the dynamics whose existence is predicted by experimental and numerical quantum spectra of twoelectron atoms such as the helium atom. 
Relevant Publications 

Other information 
Title  Electromagnetic compatibility in complex environments: predicting the propagation of electromagnetic waves using wavechaos theory 

Group(s)  Industrial and Applied Mathematics, Mathematical Physics 
Proposer(s)  Dr Stephen Creagh, Prof Gregor Tanner 
Description  The focus of this project is the development of a mathematical framework to understand the propagation of electromagnetic fields within complicated environments – a challenging task especially in the high frequency limit. Modern technology is typically stuffed with electronic componentry. Devices ranging from a mobile phone to a pc to an Airbus A380 will have many internal electronic components operating at high frequencies and therefore radiating electromagnetic waves. If the waves radiated from one component are strong enough, they can interfere with the functioning of another component somewhere else in the unit. The field of Electromagnetic Compatibility (EMC) aims to mitigate these effects by better understanding the emitted radiation. The outcome of the research will help to design electronic devices, which are protected from interference from other EM sources within buildings, pc enclosures or even planes. The innovative idea in the proposed approach rests on combining EMfield propagation with ideas of chaos theory and nonlinear dynamics. In particular, the representation of waves emitted from a complex source is described in terms of their raydynamics in phase space using the socalled Wigner distribution function (WDF) formalism. It allows us to replace the wave propagation problem with one of propagating classical densities within phase space.

Relevant Publications 

Other information 
Title  Wave propagation in complex builtup structures – tackling quasiperiodicity and inhomogeneity 

Group(s)  Industrial and Applied Mathematics, Mathematical Physics 
Proposer(s)  Prof Gregor Tanner, Dr Stephen Creagh 
Description  Computing the dynamic response of modern aerospace, automotive and civil structures can be a computationally challenging task. Characterising the structural dynamics in terms of waves in a uniform or periodic medium is often an important first step in understanding the principal propagating wave modes. Real mechanical structures are rarely fully periodic or homogeneous – variations in shape or thickness, boundaries and intersections as well as curvature destroy the perfect symmetry. The aim of the project is to extend periodic structure theory to wave propagation in quasiperiodic and inhomogeneous media such as stiffened structures. The modelling of waves can then be recast in terms of Bloch theory, which will be modified by using appropriate energy or flux conservation assumptions. The information about the propagating modes will then be implemented into modern highfrequency wave methods – such as the socalled Dynamical Energy Analysis developed in Nottingham  making it possible to compute the vibrational response of structures with arbitrary complexity at large frequencies.

Relevant Publications 

Other information 
Title  Critical random matrix ensembles 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Ossipov 
Description  In Random Matrix Theory (RMT) one deals with matrices whose entries are given by random variables. RMT has a great number of applications in physics, mathematics, engineering, finance etc. In this project, a particular class of random matrix ensembles  critical random matrix models will be studied. These models describe statistical properties of disordered systems at a point of the quantum phase transition. Using RMT one can compute various critical exponents, correlation functions and other physically important quantities. 
Relevant Publications 

Other information 
Title  Models of Quantum Geometry 

Group(s)  Mathematical Physics 
Proposer(s)  Prof John Barrett 
Description  Noncommutative geometry is a generalisation of differential geometry where the "functions" on the space are not required to commute when multiplied together. This study is based on the approach to noncommutative geometry pioneered by Alain Connes. It has a number of applications, the most spectacular being the discovery that the fields in the standard model of particle physics have the structure of a noncommutative geometry. This noncommutativity relates to the "internal space" i.e. a geometric structure at every point of spacetime, and reveals itself in the nonabelian gauge groups, the Higgs and their couplings to fermion fields. The new idea is to use the noncommutative geometry also for spacetime itself, which one hopes will eventually give a coherent explanation of the structure of spacetime at the Planck scale. There are a number of projects investigating aspects of these quantum geometry models and related mathematics. It also uses techniques from topology, algebra, category theory and geometry, as well as numerical computations. The motivation is to study models that include gravity, working towards solving the problem of quantum gravity, and to study implications for particle physics. For the latest information on this research, please see my homepage https://johnwbarrett.wordpress.com/ 
Relevant Publications 

Other information 
Title  Hydrodynamic simulations of rotating black holes 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Silke Weinfurtner 
Description  We are currently carrying out an experiment to study the effects occurring around effective horizons in an analogue gravity system. In particular, the scientific goals are to explore superradiant scattering and the black hole evaporation process. To address this issue experimentally, we utilize the analogy between waves on the surface of a stationary draining fluid/superfluid flows and the behavior of classical and quantum field excitations in the vicinity of rotating black. This project will be based at the University of Nottingham at the School of Mathematical Sciences. The two external collaborators are Prof. Josef Niemela (ICTP, Trieste in Italy) and Prof. Stefano Liberati (SISSA, Trieste in Italy). The external consultant for the experiment is Prof. Bill Unruh, who will be a regular visitor. The PhD student will be involved in all aspects of the experiments theoretical as well experimental. We require an enthusiastic graduate with a 1st class degree in Mathematics/Physics/Engineering (in exceptional circumstances a 2(i) class degree can be considered), preferably of the MMath/MSc level. Candidates would need to be keen to work in an interdisciplinary environment and interested in learning about quantum field theory in curved spacetimes, fluid dynamics, analogue gravity, and experimental techniques such as flow visualisation (i.g. Particle Imaging or Laser Doppler Velocimetry) and surface measurements (i.g. profilometry methods). 
Relevant Publications 

Other information 
Title  Gravity as a theory of connections 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Kirill Krasnov 
Description  General Relativity is normally described as a dynamical theory of spacetime metrics. However, GR is a rather complicated theory  think about the rather nontrivial exercise of deriving Schwarzschild solution, with its computation of Christoffel symbols, then the curvature tensor, then Ricci tensor. At the same time, it has been appreciated for a long time that one can simplify the GR formalism by using differential forms. Indeed, the exercise leading to Schwarzschild solution does become simpler if one uses tetrads and the spin connection instead of the metric and the affine connection. Also in 3 spacetime dimensions General Relativity is best thought of as a theory of flat Poincare connections, with the action describing the dynamics being that of ChernSimons theory. A point of view on 3D gravity as a theory of connections has been extremely successful both classically (in describing the space of all possible solutions of 3D GR) and quantum mechanically (in quantising the space of solutions and obtaining an explicit description of the arising Hilbert space). This PhD project will concern itself with developing a similar language for 4D GR. Thus, it turns out to be possible to describe 4D GR as a dynamical theory of connections rather than metrics. The metric appears as a derived notion, and is constructed in a certain way from the curvature of the connection. There are many possible projects within this general area of development. One can either explore how some concrete solutions of GR are obtained in this way, or study the quantum mechanics of gravity (i.e. perturbative quantum gravity) in this language. The language of connections is simpler in many aspects than the usual metric formalism for GR, and the hope is that this simplicity will lead to qualitatively new understanding of what gravity really is. 
Relevant Publications 

Other information 
Title  Acceleration, black holes and thermality in quantum field theory 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Jorma Louko 
Description  Hawking's 1974 prediction of black hole radiation continues to inspire the search for novel quantum phenomena associated with global properties of spacetime and with motion of observers in spacetime, as well as the search for laboratory systems that exhibit similar phenomena. At a fundamental level, a study of these phenomena provides guidance for developing theories of the quantum mechanical structure of spacetime, including the puzzle of the microphysical origin of black hole entropy. At a more practical level, a theoretical control of the phenomena may have applications in quantum information processing in situations where gravity and relative motion are significant, such as quantum communication via satellites. Specific areas for a PhD project could include: Model particle detectors as a tool for probing nonstationary quantum phenomena in spacetime, such as the onset of Hawking radiation during gravitational collapse. See arXiv:1406.2574 and references therein. Black hole structure behind the horizons as revealed by quantum field observations outside the horizons. See arXiv:1001.0124 and references therein. Quantum fields in accelerated cavities. See arXiv:1210.6772 and arXiv:1411.2948 and references therein. 
Relevant Publications 

Other information 
Title  Scattering in disordered systems with absorption: beyond the universality 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Ossipov 
Description  The study of wave scattering in quantum systems with disorder or underlying classical chaotic dynamics is essential for an understanding of many different physical systems. These include, for example, light propagation in random media, transport of electrons in quantum dots, transmission of microwaves in waveguides and cavities, and many others. An important feature of any real experiment on scattering is the presence of absorption. As the result, not all the incoming flux is either reflected or transmitted through system, but part of it is irreversibly lost in the environment. In recent years, considerable progress has been made in the study of scattering in disordered or chaotic quantum systems in the presence of absorption, see e.g Fyodorov, Savin & Sommers, (2005). However almost all results known so far are restricted by the so called "universal limit" described by the conventional Random Matrix Theory. The idea of the suggested project is to go beyond the "universal limit" and to investigate properties of the scattering matrix in lossy systems for the case of a quasionedimensional disordered waveguide. This model describes for example electron dynamics in a thick disordered wire or propagation of light or microwave radiation in a slab geometry. There are two recent advances making an analytical treatment of this problem feasible. The first one is a discovery of a kind of fluctuation dissipation relation between the properties of an open system in the presence of absorption and a certain correlation function of its closed counterpart. This can be exploited, for example, to relate statistics of scattering characteristics to eigenfunction fluctuations in closed systems (Ossipov & Fyodorov, 2005). The second one is a new analytical insight into properties of quasionedimensional disordered conductors, see Skvortsov & Ostrovsky, (2006). 
Relevant Publications 

Other information 
Title  Quantum learning for large dimensional quantum systems 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Madalin Guta 
Description  This project stems from the ongoing collaboration with Theo Kypraios and Ian Dryden (Statistics group, Nottingham), Cristina Butucea (Univesite Paris Est) and Thomas Monz and Philipp Schindler (Rainer Blatt trapped ions experimental group, University of Innsbruck). The aim is to explore and investigate new methods for learning quantum states of large dimensional quantum systems. The efficient statistical reconstruction of such states is a crucial enabling tool for current quantum engineering experiments in which multiple qubits can be controlled and prepared in exotic entangled states. However, standard estimation methods such as maximum likelihood become practically unfeasible for systems of merely 10 qubits, due to the exponential growth of the Hilbert space with the number of qubits. Therefore new methods are needed which are able to "learn" the structure of the quantum state by making use of prior information encoded in physically relevant low dimensioanal models. 
Relevant Publications 

Other information  Click here to find more information on this topic and some illustrations of different types of estimators. For more about my reasearch interests you can visit my homepage. 
Title  Feedback control of quantum dynamical systems and applications in metrology 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Madalin Guta 
Description 
The ability to manipulate, control and measure quantum systems is a central issue in Quantum Technology applications such as quantum computation, cryptography, and high precision metrology [1]. Most realistic systems interact with an environment and it is important to understand how this affects the performance of quantum protocols and how it can be used to improve it. The inputoutput theory of quantum open systems [2] offers a clear conceptual understanding of quantum dynamical systems and continuoustime measurements, and has been used extensively at interpreting experimental data in quantum optics. Mathematically, we deal with an extension of the classical filtering theory used in control engineering at estimating an unobservable signal of interest from some available noisy data [3]. This projects aims at investigating the identification and control of quantum dynamical systems in the framework of the inputoutput formalism. As an example, consider a quantum system (atom) interacting with an incoming "quantum noise" (electromagnetic field); the output fields (emitted photons) emerging from the interaction can be measured, in order to learn about the system's dynamical parameters (e.g. its hamiltonian). The goal is to find optimal system identification strategies which may involve input state preparation, output measurement design, and quantum feedback control. An interesting related question is to understand the informationdisturbance tradeoff which in the context of quantum dynamical systems becomes identificationcontrol tradeoff. The first steps in this direction were made in [4] which introduce the concept of asymptotic quantum Fisher information for "nonlinear" quantum Markov processes, and [5] which investigates system identification for linear quantum systems, using transfer functions techniques from control theory. A furhter goal is to develop genearal Central Limit theory for quantum output processes as a probablistic underpinning of the asymptotic estimation theory. Another direction is the recently found connection between dynamical phase transitions in manybody open systems and high precision metrology for dynamical parameters (see arXiv:1411.3914).

Relevant Publications 

Other information  Click here to find more information on this topic and some illustrations of different types of estimators. For more about my reasearch interests you can visit my homepage. 
Title  Quantum correlations in manybody systems 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Gerardo Adesso 
Description  The behaviour of physical systems at the microscopic scale obeys the laws of quantum mechanics. Quantum systems can share a form of quantum correlations known as entanglement, which is nowadays acknowledged as a resource for enhanced information processing. However, there are more general types of quantum correlations, beyond entanglement, that can be present in separable quantum states. This project deals with the characterisation of the nonclassicality of correlations in multipartite quantum systems. Interesting aspects of this project are the elucidation of the relationship between these more general forms of quantum correlations, as quantified e.g. by the "quantum discord", and entanglement in mixed multipartite quantum states. Another theme will be the identification of experimentally friendly schemes to engineer quantum correlations, and detect them in practical demonstrations, as well as rigorously assessing the usefulness of quantum correlations beyond entanglement as resources for nextgeneration quantum information protocols. 
Relevant Publications 

Other information 
Title  Quantum aspects of frustration in spin lattices 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Gerardo Adesso 
Description  Recently, a number of tools developed in the framework of quantum information theory have proven useful to tackle founding open questions in condensed matter physics, such as the characterization of quantum phase transitions and the scaling of correlations at critical points. Our contribution to the field dealt with a method, based on quantum informational concepts, to identify analytically factorized (unentangled) ground states in manybody spin models, which constitute an exact solution to generally nonexactly solvable models for specific values of the Hamiltonian parameters. In presence of frustration, ground state factorization is suppressed. Therefore the factorizability provides a qualitative handle on the degree of quantum frustration. This project will build on these premises and will seek for genuine signatures of quantum versus classical frustration in spin systems, a topic of great relevance for condensed matter. Frustrated quantum models may play a key role for hightemperature superconductivity and for certain biological processes. The relationship between frustration, disorder and entanglement is yet largely unexplored. 
Relevant Publications 

Other information 
Title  Quantum information with nonGaussian states 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Gerardo Adesso 
Description  Quantum information with continuous variable systems is a burgeoning area of research which has recorded astonishing theoretical and experimental successes, mainly thanks to the manipulation and exploitation of Gaussian states of light and matter. However, quite recently a number of tasks have been individuated which can not be perfectly implemented by using Gaussian states and operations only, and another set of processes is being explored where some nonGaussianity has been recognised as an advantageous ingredient to sharply improve performances of quantum communication. In this project the student will investigate the limitations of the Gaussian scenario in different contexts such as quantum communication, computation and estimation and, more generally, quantum technology. This is paralleled by recent progresses in the experimental generation of nonGaussian states, which further motivate their application in quantum information science. Special emphasis will be put on devising efficient methods to quantify the entanglement in selected classes of nonGaussian states, using techniques whose complexity is not exceedingly large compared to the usual tools (quadrature measurements, homodyne detection) which are effective for Gaussian states. 
Relevant Publications 

Other information 
Title  Developing new relativistic quantum technologies 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Ivette Fuentes 
Description  Relativistic quantum information is an emerging field which studies how to process information using quantum systems taking into account the relativistic nature of spacetime. The main aim of this PhD project is to find ways to exploit relativity to improve quantum information tasks such as teleportation and to develop new relativistic quantum technologies. Moving cavities and UnruhDewitt type detectors promise to be suitable systems for quantum information processing [1,2]. Interestingly, motion and gravity have observable effects on the quantum properties of these systems [2,3]. In this project we will find ways to implement quantum information protocols using localized systems such as cavities and detectors. We will focus on understanding how the protocols are affected by taking into account the nontrivial structure of spacetime. We will look for new protocols which exploit not only quantum but also relativistic resources for example, the nonlocal quantum correlations present in relativistic quantum fields. 
Relevant Publications 

Other information 
Title  Homotopical algebra and quantum gauge theories 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Schenkel 
Description  A problem which frequently arises in mathematics is that one would like to treat certain classes of maps as if they were isomorphisms, even though they are not in the strict sense. Examples are homotopy equivalences between topological spaces  remember the famous doughnut and coffee mug  or quasiisomorphisms between chain complexes of modules. Homotopical algebra was introduced by Quillen in the late 1960s as an abstract framework to address these and related problems. Since then it has found many important applications in algebra, topology, geometry and also in mathematical physics. In quantum field theory, homotopical algebra turns out to be essential as soon as one deals with models involving gauge symmetries. Recent results showed that quantum gauge theories do not satisfy the standard axioms of algebraic quantum field theory, hence they are not quantum field theories in this strict sense. To solve these problems, we initiated the development of a novel and promising approach called “homotopical algebraic quantum field theory”, which combines the basic concepts of algebraic quantum field theory with homotopical algebra and which is expected to be a suitable mathematical framework for quantum gauge theories. Specific areas for a PhD project could include: 1.) Examples of homotopical algebraic quantum field theory. This project is about investigating the symplectic geometry of solution "spaces" of gauge theories, which are generalised spaces called stacks, and developing new techniques for their quantisation. An important part will be to analyse localtoglobal properties (called descent) of the resulting quantum gauge theories. 2.) Operadic structure of homotopical algebraic quantum field theory. The algebraic operations in homotopical algebraic quantum field theory are expected to be captured in an abstract structure called a coloured operad. This project is about constructing this coloured operad and using it to obtain modelindependent results in homotopical algebraic quantum field theory. 
Relevant Publications 

Other information 
Title  Manybody localization in quantum spin chains and Anderson localization 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Ossipov 
Description  Properties of wave functions in manybody systems is very active topic of research in modern condensed matter theory. Quantum spin chains are very useful models for studying quantum manybody physics. They are known to exhibit complex physical behaviour such as quantum phase transitions. Recently, they have been studied intensively in the context of manybody localization. 
Relevant Publications 

Other information 
Title  Entanglement of noninteracting fermions at criticality 

Group(s)  Mathematical Physics 
Proposer(s)  Dr Alexander Ossipov 
Description  Entanglement of the ground state of manyparticle systems has recently attracted a lot of attention. For noninteracting fermions, the ground state entanglement can be calculated from the eigenvalues of the correlation matrix of the single particle wavefunctions. For this reason, the nature of the single particle wavefunctions is crucially important for understanding of the entanglement properties of a manybody system. 
Relevant Publications 

Other information 
Title  Gravity at all scales 

Group(s)  Mathematical Physics 
Proposer(s)  Prof Thomas Sotiriou 
Description  Various projects are available on the interplay between any of the following areas: quantum gravity, alternative theories of gravity, strong gravity and black holes. 
Relevant Publications 

Other information 
Title  Topological Resonances on Graphs 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  If a light wave in a resonator between two almost perfect mirrors shows resonance if the wavelength is commensurate with the distance between the two mirrors. If this condition is satisfied it will decay much slower than at other wavelengths which are not commensurate. This is one of the simplest mechanisms for a resonace in a wave system. There are other weill known mechanisms that rely on complexity and disorder. It has recently been observed that a netork of wire may have a further mechanism that leads to resonances. This mechanism relies on cycles in the network and leads to various signatures which cannot be explained using other wellknown mechanisms for resonances. In this project these signatures will be analysed in detail. 
Relevant Publications 

Other information 
Title  Quantum Chaos in Combinatorial Graphs 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Graphs consist of V vertices connected by B bonds (or edges). They are used in many branches of science as simple models for complex structures. In mathematics and physics one is strongly interested in the eigenvalues of the V x V connectivity matrix C of a graph. The matrix element C_ij of the latter is defined to be the number of bonds that connect the i'th vertex to the j'th vertex. In this PhD project the statistical properties of the connectivity spectra in (generally large) graph structures will be analysed using methods known from quantum chaos. These methods have only recently been extended to combinatorial graphs (Smilansky, 2007) and allow to represent the density of states and similar spectral functions of a graph as a sum over periodic orbits. The same methods have been applied successfully to metric graphs and quantum systems in the semiclassical regime for more than two decades. 
Relevant Publications 

Other information 
Title  Supersymmetric field theories on quantum graphs and their application to quantum chaos 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Quantum graphs are a paradigm model for quantum chaos. They consist of a system of wires along which waves can propagate. Many properties of the excitation spectrum and the spatial distribution of standing waves can be mapped exactly onto a supersymmetric field theory on the network. In a meanfield approximation one may derive various universal properties for large quantum graphs. In this project we will focus on deviations from universal behaviour for finite quantum graphs with the fieldtheoretic approach. 
Relevant Publications 

Other information 
Title  Pseudoorbit expansions in quantum graphs and their application 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Quantum graphs are a paradigm model to understand and analyse the effect of complexity on wave propagation and excitations in a network of wires. They have also been used as a paradigm model to understand topics in quantum and wave chaos where the complexity has a different origin while the mathematical framework is to a large extent analogous. Many properties of the waves that propagate through such a network can be described in terms of trajectories of a point particle that propagates through the network. The ideas is to write a property of interest as a sum over amplitudes (complex numbers) connected to all possible trajectories of the point particle. These sums remain challenging objects for explicit evaluations. Recently a numer of advanced methods for their summation have been introduced. The latter are built on socalled pseudoorbits. In this project these methods will be develloped further and applied to questions related to quantum chaos and randommatrix theory. 
Relevant Publications 

Other information 
Title  The tenfold way of symmetries in quantum mechanics. An approach using coupled spin operators. 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  About 50 years ago Wigner and Dyson proposed a threefold symmetry classification for quantum mechanical systems  these symmetry classes consisted of timereversal invariant systems with integer spin which can be described by real symmetric matrices, timereversal invariant systems with halfinteger spin which can be described by real quaternion matrices, and systems without any timereversal symmetry which are described by complex hermitian matrices. These three symmetry classes had their immediate application in the three classical Gaussian ensembles of randommatrix theory: the Gaussian orthogonal ensemble GOE, the Gaussian symplectic ensemble GSE, and the Gaussian unitary ensemble GUE. In the 1990's this classification was extended by adding charge conjugation symmetries  symmetries which relate the positive and negative part of a spectrum and which are described by anticommutators. 
Relevant Publications 

Other information 
Title  Nonlinear waves in waveguide networks 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Many wave guides (such as optical fibres) show a Kerrtype effect that leads to nonlinear wave propagation. If th wave guides are coupled at junctions then there is an additional element of complexity due to the nontrivial connectivity of wave guides. In this project the impact of the structure and topology of the network on wave propagation will be studied starting from simple geometries such as a Yjunctions (three waveguides coupled at one junction), a star (many waveguides at one junction), or a lasso (a waveguide that forms a loop and is connected at one point to a second waveguide). 
Relevant Publications 

Other information 
Title  The statistics of nodal sets in wavefunctions 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  If a membrane vibrates at one of its resonance frequencies there are certain parts of the membrane that remain still. These are called nodal points and the collection of nodal points forms the nodal set. Building on earlier work this project will look at the statistical properties of the nodal set  e.g. for 3dimensional waves the nodal set consists of a coillection of surfaces and one may ask questions about how the area of the nodal set is distributed for an ensemble of membranes or for an ensemble of different resonances of the same membrane. This project will involve a strong numerical component as wavefunctions of irregular membranes need to be found and analysed on the computer. Effective algorithms to find the area of the nodal set, or the number of domain in which the sign does not change (nodal domains) will need to be developed andimplemented. 
Relevant Publications 

Other information 
Title  Coherent states, nonhermitian Quantum Mechanics and PTsymmetry 

Group(s)  Mathematical Physics, Industrial and Applied Mathematics 
Proposer(s)  Dr Sven Gnutzmann 
Description  Heisenberg's uncertainty principle states that momentum and position cannot be sharp at the same time because there is a lower bound for the product of the uncertaincies. Coherent states can be defined as the states that minimize the uncertainty  in this sense they are as close as quantum mechanics allows to describe a classical point particle. When a quantum system starts in a coherent states it's expectation values follow the classical equations of motion while the shape of the wave function often changes only very slowly. Coherent states are an important tool to understand the corresp[ondence between quantum and classical dynamics. In this project this correspondence will be analysed for a generalized quantum dynamics where the Hamilton operator is not required to be Hermitian. Such dynamics can arise in practice as an effective description for an open quantum system with eitehr decay or gain. Accordingly the energy eigenvalues may have an imaginary part that describes the loss or gain. Recently there have also be suggestions that nonhermitian Hamilton operators could play a fundamental role in quantum mechanics if the Hamilton operator remains symmetric with respect to a combined operatyion of parity P and time reversal T. Such PTsymmetric dynamics have a balance between gain and loss which can lead to real energy eigenvalues. Classical to quantum correspondence for such systems remains an open research topic and this project will aim at getting a clear understanding of the underlying classical dynamics using coherent states as the main tool. 
Relevant Publications 

Other information 