We carry out research in Algebra, Number Theory, and Logic. We study algebraic structures and their applications to problems arising in algebraic geometry, number theory, and combinatorics. Our work in Logic focuses on intuitionism, including implications for mathematical physics.
Modern algebra studies abstract objects, and specific instances of these objects occur throughout Mathematics. Therefore, a theorem proved about an abstract algebraic object might have applications in areas as diverse as physics, computer science, or cryptography.
Our research in algebra concerns abstract objects, such as braces, Hopf algebras, loops and neofields, and their applications to other areas of Mathematics. In particular, we study how these algebraic structures can be used to create solutions of the Yang-Baxter equation, which occurs in theoretical physics (Dr. P. Truman), and to construct Latin squares with useful and interesting properties (Dr. D. Bedford).
Our work in number theory is also algebraic in nature, and focuses on the Galois module structure of algebraic integers in extensions of local or global fields. In particular, we study Hopf-Galois theory, where a Hopf algebra takes the place of the Galois group (Dr. P. Truman).
In logic, our interest is in intuitionism, a philosophy of mathematics introduced by Brouwer that rejects actual infinity and seeks to found mathematics on the concept of potentially infinite processes. Our technical work is concerned with
- the concept of a constructive proof and its role in the semantics of mathematical statements (Dr. P. Fletcher);
- the concept of a choice sequence and its use in the foundation of the theory of real numbers (J. Appleby).
Please visit the web pages individual members of staff for more details of their research, including publications.