# Available Projects

Title | Project Discription | Level | Supervisor |
---|---|---|---|

Arithmetic of conics |
The Birch and Swynnerton-Dyer conjecture is one of the million dollar problems for the 21st century as recognized by the Clay Institute. The conjecture concerns solutions of elliptic (genus 1) curves. Recently an analogy has been proposed for the much simpler case of genus 0 curves (conics).... |
PhD Project | Dr Victor Scharaschkin |

Arithmetic of higher genus curves |
Curves of genus 0 and 1 are relatively well understood. Starting in the 1990s explicit methods have been developed for the first time to deal with more complicated curves and determine all of their rational points. Open problems in this field include the role of an object called the Brauer-Manin... |
PhD Project | Dr Victor Scharaschkin |

Euclidean rings |
Euclidean rings are algebraic structures generalizing the set of integers. Like the integers they have a division algorithm and unique factorization. Historically it has proved very difficult to determine if a ring is Euclidean or not but there have been several recent breakthroughs which are... |
PhD Project | Dr Victor Scharaschkin |

Computational number theory |
There is also scope for work in computational number theory such as primality testing and factorization methods |
PhD Project | Dr Victor Scharaschkin |

Neoclassical Invariant Theory |
Using invariant theoretic approaches, you can study various geometric and combinatorial properties of finite-dimensional representations of the classical groups. A number of projects are available. Recent work in this area includes the study of flag/Grassmann varieties, toric degenerations,... |
PhD Project | Dr Sangjib Kim |

The Positive Realization Problem |
The Positive Realization Problem deals with finding descriptions of linear systems based on non-negative matrices. It has applications in control, applied probability and statistics, yet is linear-... |
Masters Project Honours Project |
Dr Yoni Nazarathy |

Affine algebras and Langlands program |
The Langlands program is one of the most ambitious research projects in mathematics. In the past decade it has become clear that representations of affine... |
PhD Project | Dr Masoud Kamgarpour |

Root systems and their applications in representation theory |
Root systems are one of the most remarkable structures elucidated in 20th century mathematics. They have a simple definition in terms of linear algebra and combinatorics, but have... |
Honours Project | Dr Masoud Kamgarpour |

Representation theory of infinite-dimensional Lie algebras |
Conformal field theory plays a fundamental role in string theory and in the description of phase transitions in statistical mechanics. The basic symmetries of a conformal field theory are generated by infinite-dimensional Lie... |
PhD Project Masters Project |
Dr Jorgen Rasmussen |

Hitchin's fibration and its application in mathematics and physics |
Defined by the eminant mathematical physics Nigel Hitchin, the Hitchin's... |
PhD Project | Dr Masoud Kamgarpour |

Representations of jet Lie algebras |
Representations of semisimple Lie algebras is one of the most beautiful parts of mathematics in the 20th century. It draws on... |
Masters Project | Dr Masoud Kamgarpour |

Game theory and algebraic geometry |
Nash's equilibrium is a celebrated result in mathematics and economics. It is one of the foundational results in game theory. The aim of this project is to explore the relationship between game theory and algebraic geometry. |
Honours Project | Dr Masoud Kamgarpour |

Computing with Quantum Groups |
Quantum groups are remarkable q-analogues of the theory of Lie groups and Lie algebras introduced by Jimbo and... |
Masters Project Honours Project Summer Project |
Dr Peter McNamara |

Spherical Functions |
A classical result due to MacDonald shows that certain functions (so-called spherical functions) that appear in representation theory of p-adic groups are well known symmetric functions. This project will study this theorem of MacDonald,... |
Masters Project Honours Project |
Dr Peter McNamara |

KLR algebras/categorified quantum groups |
In 2008, Khovanov and Lauda introduced a remarkable family of algebras designed for the... |
PhD Project Masters Project |
Dr Peter McNamara |