Department of Mathematics
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Department of Mathematics

Potential PhD projects

We invite high-calibre students with a passion for research to join us and study for a PhD. Potential PhD topics are outlined below. Find out more about applying to a research degree in the Department of Mathematics.


Quantifying and Modelling the Cryptocurrency Ecosystem

Supervisor: Dr Andrea Baronchelli

Bitcoin and the other cryptocurrencies have gained growing attention for the past years and are today a major financial and economic reality. Strikingly, however, not much is known about the behaviour of the cryptocurrency market or the dynamics occurring on the transaction networks of single cryptocurrencies. This project aims to fill this knowledge gap by adopting a complex systems perspective, combining Data Science approaches with concepts and tools from Network Science and mathematical modelling.


Quantifying human behaviour - From Data to Models

Supervisor: Dr Andrea Baronchelli

The abundance of digital traces has opened unprecedented possibilities to gain a quantitative insight into human dynamics. From how we move in cities to how we shop or communicate online, Data Science and quantitative modelling have shed light into several behaviours that had been so far studied only from a qualitative point of view. The candidate for this project will select one specific problem among those mentioned above and tackle it combining Data Science approaches with concepts and tools from Network Science and mathematical modelling, in a project that will possibly involve collaborations with experts from other disciplines.


Modelling evolution in structured populations using multiplayer games

Supervisor: Professor Mark Broom

Evolutionary games play an important role in the modelling of the evolution of biological populations. The classical models of evolution have been developed to incorporate structured populations using evolutionary graph theory and, more recently, a new framework has been developed to allow for more flexible population structures which potentially change through time and can accommodate multiplayer games with variable group sizes. In this project we will focus on two major problems that have so far not been addressed. Firstly we will systematically find analytical solutions where possible for the simplest models. Secondly we will develop a range of approximation methods to model large complex populations. The latter work will involve both abstract models and models representing specific real scenarios.


Game on dynamically evolving networks

Supervisor: Professor Mark Broom

Animal (and human) populations contain a finite number of individuals with social and geographical relationships which evolve over time, at least in part dependent upon the actions of members of the population. These actions are often not random, but chosen strategically, and recent work has introduced a game-theoretical model of a population where the individuals have an optimal level of social engagement, and form or break social relationships strategically to obtain the correct level. In this project one focus will be on developing model strategy sets appropriate to different player capabilities, representing different animal species. As this is a newly developed model, there are a number of open problems and potential directions that the project can take depending upon the interests and skills of the student.


Measures of Entanglement in Excited States of Quantum Field Theory

Supervisor: Dr Olalla Castro-Alvaredo

The mathematical quantification of entanglement in quantum systems is an extremely active area of current research. At the present time, there is good understanding of various measures of entanglement for the ground state of wide variety of quantum systems but less is know about the features of entanglement in excited states. The aim of this project is study universal features of the entanglement entropy (a particular measure of entanglement) in particular classes of excited states. We want to do this by employing a quantum field theory approach known as the branch point twist field approach which has proven very successful in understanding features of entanglement in the ground state. The aim of the project is to extend this approach to excited states and potentially to various measures of entanglement.


Logarithmic Negativity of Quantum Spin Chains with Highly Degenerate Ground States

Supervisor: Dr Olalla Castro-Alvaredo

The mathematical quantification of entanglement in quantum systems is an extremely active area of current research. At the present time there is good understanding of various measures of entanglement for the ground state of wide variety of quantum systems but less is know about the features of entanglement when the ground state is highly degenerate. Previous work involving this project's supervisor has shown that quantum systems exhibiting this feature also exhibit new entanglement behaviours, in particular when their entanglement entropy (a particular measure of entanglement) is studied. So far no other measure of entanglement has been studied in this context and it is expected that other measures will display interesting novel features too. The aim of this project will be to study a measure of entanglement known as the logarithmic negativity by employing an approach based on local twist operators. Intriguingly, similar entanglement properties have been found for random quantum spin chains, even though their relationship with highly degenerate ground state systems is unclear. The project will also try to clarify whether or not there is a connection between these two types of quantum systems.


Modular representation theory of the symmetric group via the partition algebra

Supervisors: Dr Anton Cox, Dr Maud De Visscher.

This project aims to advance our knowledge of the representation theory of the symmetric group via its connections to the partition algebra. Over the complex numbers, the representation theory of the symmetric group is semisimple and the simple modules have been classified and explicitly constructed by Young early in the last century. However, over fields of positive characteristic, the situation is considerably more complicated. A classification of simple modules exists but constructing the corresponding simple modules is a major open problem. Moreover, the representation theory in this case is no longer semisimple. So we have many other open questions such as understanding its cohomology.

The partition algebra was introduced by Martin in the early 90’s in the context of Statistical Mechanics. It has strong connections with the symmetric group which haven’t been fully explored to date, especially over fields of positive characteristic. In this project, you will develop these links further and investigate the implications on the representation of the symmetric group and in particular on its cohomology.


Representation theory of diagram algebras

Supervisors: Dr Anton Cox, Dr Maud De Visscher

Diagram algebras have played an increasingly important role in recent developments in the representation theory of Lie algebras and their generalisations. A number of these developments were pioneered at City, in the work of Cox and De Visscher, and this PhD project will continue the study of such algebras and their applications. There are a number of possible research directions, both in the classical area of Brauer algebras and their generalisations and in the more recently defined study of diagrammatic Cherednik algebras, which will involve studying the representation theory of such algebras over fields of positive characteristic.


Entropic uncertainty for quantum theories on noncommutative space-time

Supervisor: Professor Andreas Fring

The main aim of this proposal to generalize Heisenberg’s uncertainty relations from two canonical variables to three canonical variables. The study is motivated by the fact that in noncommutative space-time structures measurements of such type naturally occur. The project will investigate the relations on various types of these spaces, especially including those involving minimal lengths. Emphasis will be placed on the construction of squeezed states that minimize the inequalities. Star products for the noncommutative spaces will be constructed with threefold symmetries together with their operator realisations. The findings will be tested for Schrödinger-Robertson and entropic uncertainty relations.


Integrable quantum field theories based on Lorentzian Kac–Moody algebras

Supervisor: Professor Andreas Fring

Classical and quantum integrable systems based on Lie algebras and Kac–Moody algebras have been well studies and understood. Examples of these are Toda field theories and models of Calogero type. Recently it has been discovered that generalizations of these algebras, hyperbolic Kac–Moody algebras, play an important role in M-theory as they represent their larger symmetry. The purpose of these project is to formulate new integrable systems based on these type of algebras and investigate their classical and quantum field theoretical properties.


Interactions between Gauge/String Theories, Algebraic Geometry and Number Theory

Supervisor: Professor Yang-Hui He

The project is on the interface between gauge theory and pure mathematics. In particular, there is an active field of study, originally inspired by string theory, on the geometry of compact and non-compact algebraic varieties, especially Calabi-Yau manifolds, and manifestations in quantum field theory. Specifically, quiver representations provide a fascinating link between the gauge theory and the combinatorics and geometry of the moduli space.

The reader is referred, for example, to review articles:

  • Y.-H. He, ``Calabi-Yau Varieties: from Quiver Representations to Dessins d'Enfants,'' arXiv:1611.09398
  • Y.-H. He, ``Calabi-Yau Geometries: Algorithms, Databases, and Physics,'' arXiv:1308.0186

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