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Structure and Symmetry

Boundary States in integrable quantum spin chains with open boundaries

Reference Number: 2017-HW-Maths-07

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.

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Calibrated Submanifolds

Reference Number: 2017-HW-Maths-42

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.

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Category theory, Entropy and diversity, Magnitude of metric spaces, Magnitude homology, Integral geometry

Abstract: See the link below.

Description of available projects.

Classical Structures of the (2,0)-Theory

Reference Number: 2017-HW-Maths-40

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.

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Complete Manifolds with Special Holonomy

Reference Number: 2017-HW-Maths-42

Abstract: Manifolds with special holonomy are fundamental objects in differential  geometry: they are Einstein manifolds (i.e. they solve the Riemannian analogue of Einstein’s equations of General Relativity), carry a rich geometric structure (for example they support interesting minimal submanifolds and gauge theoretic equations) and play a key role in String and M theory. The overall aim of this project is the study of certain classes of complete non-compact manifolds with special holonomy, in particular 8-­‐dimensional manifolds with Spin(7) holonomy. Spin(7) manifolds are not easy to construct and one of the aims of the project is to explore constructions of Spin(7) manifolds from lower-dimensional better understood geometries. The project has direct connections with hyperkähler and Calabi-­‐Yau geometry, draws inspiration from theoretical physics, and involves the use of analytic tools to work on non-­compact and singular manifolds.

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Counting geodesics in groups

Reference Number: 2017-HW-Maths-05

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.

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Evolution problems in non-self-adjoint spectral theory

Reference Number: 2017-HW-Maths-37

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.

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Reference Number: 2017-HW-Maths-45

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.

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Geometry of T-duality

Reference Number: 2017-HW-Maths-39

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.

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Higher Quantisation of Spaces

Reference Number: 2017-HW-Maths-42

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.

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Integrable quantum spin chains with open boundaries

Reference Number: 2017-HW-Maths-06

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.

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Knot homology and supersymmetric gauge theory

Reference Number: 2017-HW-Maths-23

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.

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Non-commutative geometry and quantum symmetries in 3d quantum gravity

Reference Number: 2017-HW-Maths-20

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.

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Numerical approximation of rigorous enclosures for the spectrum of J-self-adjoint operators

Reference Number: 2017-HW-Maths-36

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.

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Quantisation of Higher Gauge Theories

Reference Number: 2017-HW-Maths-41

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.

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Renormalisation interfaces in two-dimensional quantum field theory

Reference Number: 2017-HW-Maths-08

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.

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Solving equations in groups

Reference Number: 2017-HW-Maths-04

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.

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Spectral Pollution

Reference Number: 2017-HW-Maths-35

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.

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Spin-orbit force in the Skyrme model

Reference Number: 2017-HW-Maths-21

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.

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Supersymmetric gauge theory and quantum Hitchin systems

Reference Number: 2017-HW-Maths-24

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.

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SUSY and the ASEP

Reference Number: 2017-HW-Maths-44

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.

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Synthetic topological gauge fields in three dimensions

Reference Number: 2017-HW-Maths-22

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.

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The Geometry of Abelian 3-folds via Stability Conditions

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.

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Wall-crossing, stability conditions and moduli spaces

Abstract: "Stability conditions" and "wall-crossing" are abstract categorical concepts that were originally invented in order to give rigorous mathematical definitions of concepts in string theory. But recent breakthroughs have allowed us to apply then in order to answer fundamental basic questions purely within algebraic geometry. The specific direction of the project would depend very much on the taste and preferences of the PhD student. One aim could be to extend such applications from the case of a single variety (a space defined by a system of polynomial equations) to a family of varieties (defined by a system of  polynomial equations with parameters). The abstract machinery is now in place in the setting of families, and is asking to be applied to give us a better understanding of algebraic varieties in families.

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