Structure and Symmetry

Abstract: Geometric group theory is the field of mathematics trying to understand groups through their actions on metric spaces. Given a group, there is a large spectrum of possible actions, from actions on Cayley graphs of the group to actions on more complicated complexes. Often, a lot of information can be derived from the action when the space acted upon satisfies a mild form of non-positive curvature. Such actions in non-positive curvature have been shown to exist for large classes of groups: graph products of groups, fundamental groups of 3-manifolds, certain Artin groups, etc. When the action possesses strong geometric and dynamical properties reminiscent of hyperbolic geometry, one can hope to obtain a finer understanding of the group: determining whether it satisfies the `Tits alternative' (an important dichotomy about its subgroups), describing its automorphism group, etc. The goal of this project will be to investigate the structure (subgroups, endomorphisms, etc.) of several classes of groups from such a geometric perspective. This project brings together group theory, dynamics in non-positive curvature, and combinatorial geometry.

Abstract: Exact  expressions  for boundary  states  of  integrable quantum spin
chains  (eigenstates  of  the   Hamiltonian  acting  parallel  to the
boundary) can sometimes  be obtained via the  vertex operator approach
to solvable lattice models developed  by Jimbo, Miwa and collaborators
in the  1990s. This  approach relies on  the representation theory of
quantum  affine algebras.   In this  project you  will construct such
boundary  states in  a new  class of  models -  those associated with
'twisted Yangian' algebraic symmetries.   You will analyse the scaling
limit of these  models and compare your results  with calculations for
the corresponding  boundary scattering matrices of  integrable quantum
field  theories. You  will then  use your  boundary states  in order  to
construct exact  expressions for  correlation functions  in integrable
quantum spin chains with open boundaries.

Abstract: Calibrated submanifolds form a special class of volume minimising submanifolds. While minimal submanifolds are critical points of the area functional and are solutions of a second-­‐order PDE, the calibrated condition defines a constraint on the tangent space at every point of the submanifold (in other words, a first-­‐order PDE) that implies the second order minimal submanifold equation and the much stronger condition of being an absolute minimum of the area functional. Standard examples of calibrated submanifolds are complex submanifolds of Kähler manifolds, e.g. submanifolds of complex projective space cut-­out by homogeneous polynomial equations. More in general, calibrated submanifolds of so-‐called manifolds with special holonomy play a key role in geometric and physical applications: for example, “counting” calibrated submanifolds in a given homology class is expected to lead to invariants of Riemannian manifolds with special holonomy.  
In this project we aim to construct and study classes of calibrated submanifolds in complete non-­‐compact manifolds with special holonomy. We will focus on  cases where symmetries of the ambient space allow one to reduce the problem  to a PDE system in low dimensions (sometimes even to an ODE system), albeit a  degenerate/singular one due to fixed points for the action of the symmetry  group. The project combines geometric and physical motivations and ideas with  analytic aspects such as free boundary value problems in PDEs, the study of  degenerate PDEs and geometric measure theory.

Abstract: The (2,0)-theory is a highly supersymmetric six-dimensional superconformal field theory which is rather mysterious: while its existence was postulated as early as 1995, the theory is still rather poorly understood. It is a widely held belief that the (2,0)-theory only exists on the quantum level. Therefore, many current research projects are devoted to studying its quantum aspects. Recent results, however, suggests that there might be a classical description of this theory within the framework known as higher or categorified gauge theory. In this project, we will study to what extent this classical description exists and fits the expectations from string theory. For this, some background in string theory is required. Knowledge in differential geometry and category theory is useful.

Abstract: To each finitely generated group one can attach the Cayley graph: a graph whose vertices are the group elements, and an edge connects two vertices if these are related via multiplication by a generator. If one counts all the geodesic, or shortest, paths, between the identity element/vertex and vertices at distance n from the identity, then one is looking at the geodesic growth function of the group. Much is known about this function, but a lot is also left to explore. For example, does there exist a group where this function is algebraic, but not rational? Or does there exist a group where this function is bigger than polynomial, but less than exponential? This project brings together group theory, combinatorics, formal languages, and computational experiments.

Abstract: Operators underlying non-linear phenomena are often non-self-adjoint. As an example, linearising the equation governing the behaviour of a viscous fluid inside a rotating cylinder with axis parallel to the ground, gives rise to a highly non-self-adjoint operator with boundary and interior singularities. The evolution equation associated to this operator exhibits very unusual properties: there is a dense set of initial conditions (the eigenfunctions of the operator and any finite linear combination of them) for which there is a solution for all time, yet there is also a dense set of initial conditions for which there is no solution for any positive time. Many more examples of this sort have been found recently. The goal of this PhD project will be to closely examine specific classes of non-self-adjoint operators for which the associated evolution problem exhibits a wild behaviour.

Abstract: T-duality is a fascinating symmetry that distinguishes string theory from quantum field theories of point particles. Locally, the underlying geometry is described in terms of categorified Lie algebroids but many of the global aspects are still poorly understood. It is the aim of this project to study these global aspects in detail and to identify the right categorified geometrical structures for their description. These geometrical structures will then be applied in the study of Double Field Theory, which is a T-duality invariant formulation of the low-energy sector of string theory, as well as to Exceptional Field Theory, the U-duality invariant lift of Double Field Theory to M-theory. Given the content of the project, some background in differential geometry, category theory and string theory is helpful.

Abstract: Integrable quantum  field theories and integrable  quantum spin chains
with  boundaries have  been  a topic  of major  interest  for over  20
years. Applications arise  in both string theory  and condensed matter
physics. The word integrable means that these systems possess enhanced
symmetries - with the consequence that some of their properties can be
computed  exactly.  In particular,  it  is  in principle possible  to
compute their energy eigenvalues  exactly. These eigenvalues are given
in terms of the solution of a system of equations called 'Bethe ansatz
equations',  which in  term come  from  a more  fundamental system  of
equations called  'Baxter relations'. Baxter relations  are difference
equations for a polynomial Q(z). The  modern  construction and  understanding  of  quantum spin chains
relies on  the representation theory  of quantum groups, also know as
quasi-triangular Hopf  algebras. While this picture  is well-developed
for closed,  periodic quantum  spin chains, it  is only  very recently
that Baxter  relations have  been fully  understood in  this language.
The main goal  of this project is to develop  a parallel understanding
of Baxter  relations in 'open'  quantum spin  chains - that  is, those
with two independent integrable boundary conditions.

Abstract: Quantization of locally non-geometric closed string backgrounds reveals a nonassociative deformation of phase space geometry. Nonassociative differential geometry has been developing over the last few years, but there are many aspects still to be understood, particularly its metric aspects. This project concerns the development of nonassociative geometry from various points of view, including quasi-Hopf twist deformation techniques and L-infinity algebras, with an eye to understanding and developing a nonassociative theory of gravity as an effective low-energy theory of closed strings in non-geometric backgrounds.

Abstract: The aim of this PhD project is to device strategies for computing rigorous sharp bounds/enclosures for the spectrum of J-self-adjoint operators by means of projected space methods (Galerkin methods). The theory of computation for spectra of self-adjoint operators is classical and well developed. J-self-adjoint operators share many properties with their self-adjoint counterpart however a systematic procedure for computing their spectrum remains as a largely unsolved open problem. During this PhD, the student will examine general strategies for numerically estimating spectra of J-self-adjoint operators. Particular directions will involve the following. (a) Computation of enclosures of spectra for operators arising in the theory of rotating viscous fluids. (b) Computation of sharp enclosures for the complex isotonic harmonic oscillator. (c) Impact in the study of time evolution problems for J-self-adjoint operators.

Abstract: Please see the link below for up to date list of projects in mathematical physics at the University of Edinburgh.

Abstract: Renormalisation group (RG) is as fundamental concept in Quantum Field Theory (QFT) that describes how the physics changes under the change of energy scale. A typical renormalisation group trajectory starts from one fixed point and drives the theory to a different fixed point. In two dimensions the fixed points are described by conformal field theories that possess an infinite-dimensional symmetry algebra. While a lot in known about the end points of renormalisation group flows very little is known about the global structure of the space of flows linking the end points. Recently a new object called Renormalisation domain wall or renormalisation interface has been invented. In two dimensions this object is a line of interface between two different conformal field theories on each side. The project involves studying such objects for concrete  RG flows both analytically and numerically. The aims are to learn how to construct such objects, what information they encode about RG flows and how could they be used in gaining control over the space of flows. Although not limited to, this project involves some degree of programming and numerical computations. Some references: arXiv:1201.0767; arXiv:1211.3665; arXiv:1407.6444.

Abstract: This project is about finding certified bounds for eigenvalues in regimes where the classical methods fail catastrophically. The Galerkin method is among the most reliable tools for finding upper bounds for the eigenvalues of selfadjoint operators. It is widely used in applications, ranging from civil and mechanical engineering to thermodynamics and non-relativistic quantum chemistry. Unfortunately the Galerkin method can fail dramatically when applied to the “wrong” eigenvalue problem. This phenomenon, called spectral pollution, is notoriously difficult to predict and it arises in relativistic quantum mechanics, solid state physics and electromagnetism. The purpose of this PhD project, is to characterise on a mathematically rigorous level what spectral pollution is and then examine analytical and numerical techniques for avoiding it. These include the quadratic method, the ZMG method, the factorisation method and the perturbation method. At the end of the project the student will have a broad view on the state-of-the-art in analytical and computational spectral theory with a specific emphasis in the current topic of spectral pollution.

Abstract: In recent years much progress has been made in understanding of supersymmetric gauge theories. Exact computations have revealed interesting links with quantum integrable systems. In this PhD project we will deepen this connection and employ it to study quantum Hitchin systems.

Abstract: There has recently been rapid progress in experimental techniques for realising gauge fields with interesting topological properties by manipulating  cold atom systems with Laser beams.  It is known how to produce   certain magnetic fields with non-trivial linking numbers, but  there is no general understanding of  the knotted and linked structures which one can produce in practice. The goal of this project is to develop a systematic theory of the laser beams required to produce a given link or knot.