Structure and Symmetry

Abstract: Recently there has been a renewed interest in studying the general consequences of conformal symmetry in Quantum Field Theory. Conformal symmetry imposes an infinite number of relations. The programme of solving such general relations is called conformal bootstrap. The project proposes to study futher study general consequences of modular invariance and crossing symmetry for two-dimensional confromal field theories (2D CFT's) with and without a boundary. The aim is to unerstand better the space of 2d CFT's and establish bounds on fundamental quantities. Although not limited to, this project involves some degree of programming and numerical computations. Some references: arXiv:1111.2115; arXiv:1206.5395; arXiv:1305.2122.

Abstract: Classical instantons can be viewed as special holomorphic sheaves on complex projective 3-space. These are special subspaces of Gieseker semistable sheaves which in turn appear as limiting spaces of Bridgeland stable moduli spaces. Recent advances in the field now makes it possible to classify the moduli spaces and to use the walls to give explicit constructions of the spaces. This project would aim to carry out the construction for the instanton moduli spaces.
 

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: 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: Gauge/gravity duality implies that there are interacting states of matter whose simplest mathematical description is as a theory of gravity. The obvious differences between gravitational physics and the physics of familiar states of matter (e.g. Fermi liquids) make this counterintuitive but also advantageous. It means that theories of gravity provide mathematically consistent descriptions of new phases of quantum matter with unconventional properties, which are potentially useful toy models for some of the strange states of matter observed in real materials. Broadly, this project will involve studying the dynamics of black holes, and the implications for the transport of charge and energy in these unusual states.

Abstract: Renormalisation group flows describe how quantum field theories (QFTs) change when we change the scale of interaction. The coupling constants then change according to their beta functions that can be considered as a vector field on the space of theories. More geometry on the space of theories arises when one takes into account stress-energy tensor conservation and anomaly consistency conditions. The crucial equation describing the local geometry is a gradient formula for the beta functions. Roughly speaking it expresses the beta function as a gradient of some potential function plus additional terms related to various local tensor fields. In string theory such equations for two-dimensional QFTs can be considered as space-time equations of motion for the string fields. The project includes studying the gradient formulae, local geometry and its string theory interpretation. Possible concrete problems include N=2 supersymmetric and PT-symmetric gradient formula for boundary RG flows in two dimensions. Some references: arXiv:1310.4185, arXiv:0910.3109, arXiv:hep-th/0312197.

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 defect 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. Some references: arXiv:1201.0767; arXiv:1211.3665; arXiv:1407.6444, arXiv:1610.07489.

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.

Abstract: Bridgeland stability conditions are a new powerful tool to understand the category of coherent sheaves on varieties. Enough technology now exists for stability conditions on 3-folds of Picard rank 1 to be able to answer questions about the geometry of the underlying variety. For abelian 3-folds we also have a powerful tool in the form of the Fourier-Mukai transform which interacts well with stability conditions. This project would aim to extend the methods used to understand the geometry of abelian surfaces to 3-folds.