Algebra is the study of abstract mathematical structures that generalise well-known number systems such as the integers or the reals, and their arithmetic operations such as addition and multiplication. Some of the most important algebraic structures are rings (such as the integers), fields (such as the reals) and groups (such as the symmetries of the cube). The study of algebra is of fundamental importance to our understanding of the natural world. The electromagnetic and nuclear forces, for example, are best understood in terms of the so-called three unitary groups SU(3), SU(2) and U(1).

Traditionally, Number Theory referred to the study of the integers. However, in modern mathematics number theorists are interested in the properties of much broader classes of numbers. Whilst considered an area of pure mathematics, Number Theory is pervasive in every-day life, with prime numbers forming the key to secure e-shopping and internet banking. Famous unsolved problems in number theory are the Goldbach conjecture—the claim that every even integer greater than 2 can be written as the sum of two prime numbers—and the Riemann hypothesis, one of the million dollar millenium problems.

Available Projects

Lie algebras, Lie groups, and their representation theories are instrumental in our description of symmetries in physics and elsewhere. They also occupy a central place in pure mathematics where they often provide a bridge between...

Associate Professor Jorgen Rasmussen

In 2008, Khovanov and Lauda introduced a remarkable family of algebras designed for the...

Dr Peter McNamara

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,...

Dr Peter McNamara

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. 

Dr Masoud Kamgarpour

Representations of semisimple Lie algebras is one of the most beautiful parts of mathematics in the 20th century. It draws on...

Dr Masoud Kamgarpour

Defined by the eminant mathematical physics Nigel Hitchin, the Hitchin's...

Dr Masoud Kamgarpour

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...

Associate Professor Jorgen Rasmussen

Diagram algebras offer an intriguing mathematical environment where computations are performed by diagrammatic manipulations. Applications include knot theory and lattice...

Associate Professor Jorgen Rasmussen

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...

Dr Masoud Kamgarpour

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...

Dr Masoud Kamgarpour

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-...

Dr Yoni Nazarathy

There is also scope for work in computational number theory such as primality testing and factorization methods

Dr Victor Scharaschkin

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...

Dr Victor Scharaschkin

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...

Dr Victor Scharaschkin

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)....

Dr Victor Scharaschkin