Structure and Symmetry

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: There has been considerable recent interest in topological phases of matter from both mathematicians and physicists. The 2016 Nobel Prize in Physics was awarded to the theoretical physicists David Thouless, Duncan Haldane, and Michael Kosterlitz whose work established the role of topology in understanding various exotic forms of matter.  Gauge theories with local symmetries appear to provide a mechanism for understanding
some topological phases, including quantum Hall states and quantum spin liquids. This project will investigate theories with so-called subsystem symmetries, intermediate between gauge theories and theories with global symmetries, which are related via a quantum duality to  systems with fracton topological order. Such systems exhibit fractional excitations termed “fractons" which can either be immobile or only free to move in restricted subspaces. The phase structure of the associated classical spin models, typically containing multi-spin interactions,  will also be of interest.

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: 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: 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.