**Reference Number:** 2017-HW-Maths-07

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.

**Reference Number:** 2017-HW-Maths-42

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.

**Reference Number:** 2017-HW-Maths-40

**Reference Number:** 2017-HW-Maths-42

**Reference Number:** 2017-HW-Maths-05

**Reference Number:** 2017-HW-Maths-37

**Reference Number:** 2017-HW-Maths-45

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.

**Reference Number:** 2017-HW-Maths-39

**Reference Number:** 2017-HW-Maths-42

**Reference Number:** 2017-HW-Maths-06

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.

**Reference Number:** 2017-HW-Maths-23

**Reference Number:** 2017-HW-Maths-20

**Reference Number:** 2017-HW-Maths-36

**Reference Number:** 2017-HW-Maths-41

**Reference Number:** 2017-HW-Maths-08

**Reference Number:** 2017-HW-Maths-04

**Reference Number:** 2017-HW-Maths-35

**Reference Number:** 2017-HW-Maths-21

**Reference Number:** 2017-HW-Maths-24

**Reference Number:** 2017-HW-Maths-44

**Reference Number:** 2017-HW-Maths-22