Structure and Symmetry

Abstract: Anomalies in various physical theories have been understood from the relatively new perspective of functorial field theories, which provide a rigorous approach to quantum field theory, and have broad applications to string theory (for instance, in the 6-dimensional (2,0)-theory) and to condensed matter physics (for instance, in symmetry-protected topological phases of matter). This project will study various topical features of anomalies and other field theoretic phenomena from the functorial perspective in this rapidly-evolving field.

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: Already Bernhard Riemann suggested that at some microscopic level, our idea of space as a smooth manifold should be replaced by something more general. Since the invention of quantum mechanics, we know that geometrically quantised manifolds are a good candidate for such a generalisation. In certain models inspired by string theory, these quantised spaces can indeed arise and evolve dynamically, which provides us with an interesting mechanism for the creation of space and the origin of General Relativity. A more detailed study, however, suggests that ordinary geometric quantisation is not sufficient, but has to be extended to a categorified notion of geometric quantisation, also known as higher quantisation. So far, higher quantisation has been developed only partially. In this project, we will study new approaches to this problem, using inspiration from string theory and categorified geometry. This should significantly extend our idea of what spacetime is at a very fundamental level. We will also study dynamical principles for such quantised spacetimes and compare the results to those of General Relativity. A good background in differential geometry is required. Some knowledge of category theory is useful.

Abstract: Physical questions in supersymmetric gauge theories often turn out to reveal deep connections with mathematics. One particularly interesting theme in this respect is knot theory. In this PhD project we will study how knot homologies make their entrance into supersymmetric gauge theory and string theory.

Abstract: In three dimensions, Einstein's theory of gravity can be formulated as a Chern-Simons theory and quantised  using techniques from Poisson-Lie  and quantum group theory. This provides a natural setting for exploring the link between quantum gravity, non-commutative spacetimes and deformed symmetries. The theory is now well-understood in the case of vanishing cosmological constant, and the goal of this project is to look at the case of negative cosmological constant. In this case, the classical theory contains BTZ black holes, and one of the specific goals is to understand their quantum analogues. Recently, interesting links have been established between quantised Chern-Simons theories and the so-called Kitaev model in topological quantum computing, and this may allow new perspectives on this problem.

Abstract: Please see the link below for up to date list of projects in Algebra, Geometry and Representation Theory at the University of Edinburgh.

Abstract: Higher gauge theories are categorified generalisations of ordinary gauge theories such as Yang-Mills theory or Chern-Simons theory. They are expected to play an important role within string theory and string field theory. We have some understanding of these theories at the classical level, but their quantisation has not been studied yet in any reasonable detail. It is the aim of this project to fill this gap. In particular, we want to translate many of the interesting results in quantum Yang-Mills theory and quantum Chern-Simons theory to the higher analogues of this gauge theories, with a focus on topological invariants. Some knowledge of quantum field theory is required. Further knowledge in differential geometry and category theory is useful.

Abstract: Imagine an equation of the form XaYYbZZc=1 in a group G, where X,Y, Z are variables and a, b, c some elements in G. Does this equation have solutions, and if it does, what are they? The answer depends very much on the group, whether it is free, hyperbolic, nilpotent or some other type. In some cases these questions, for arbitrary equations, are unsolvable, in other cases they are well understood but quite difficult. This project would revolve around understanding equations in nilpotent groups, and the base case would be the 3x3 Heisenberg group, where very little is known in terms of describing the solutions to an equation. Alternatively, depending on the background of the applicant, it could involve equations in some groups acting on rooted trees, such as the Grigorchuk group. This project brings together group theory, combinatorics, computational complexity, and possibly some algebraic geometry and formal languages, and it can be treated theoretically or rather computationally.

Abstract: In the Skyrme model, nucleons are modeled by topological solitons in the Skyrme model, also called Skyrmions. It is now well-understood that the interactions of topological solitons are governed by both static and kinematic effects. The latter are related to the non-trivial Riemannian geometry of the configuration space (or moduli space) of solitons. The goal of this project is to understand  how  spin-orbit forces, which play a crucial role in nuclear binding,  are  described in the Skyrme model. Since just forces are quadratic in orbital speeds, it is  natural to interpret them  in terms of kinematic effects. The goal of this project is to do this in practice.  The work would combine geometry  and physics and will probably also include some numerical work.

Abstract: Supersymmetry (SUSY), a symmetry between bosons and fermions, has been conspicuous by its absence in particle physics where it was originally formulated. However, an exact lattice supersymmetry has been found in various 1D spin chains where it relates chains of different lengths. The asymmetric exclusion process (ASEP) is known to be closely related to the XXZ spin chain and it has already been observed that a transfer matrix ansatz (TMA) for the ASEP exists which satisfies a relation that takes a similar form to the defining relation of lattice supersymmetry. The project will investigate the implications of this for the solvability and spectrum of the ASEP.

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.