Discrete Mathematics
We study Mathematical structures arising from algebra, combinatorics, geometry, and number theory. Our work has connections and applications to theoretical computer science, theoretical physics, and many other areas of mathematics.
Discrete objects such as sets, graphs, permutations, groups, rings, and fields occur throughout mathematics. Research on these objects has applications in subjects including physics, theoretical computer science, information theory, finance, and cryptography.
Our work in algebra focusses on Hopf-Galois theory and the theory of skew braces.
Skew braces are mathematical objects that provide an algebraic framework for studying set theoretic solutions of the Yang-Baxter equation, which is one of the fundamental equations of theoretical physics. We study the structure and properties of skew braces and the associated solutions, their connections with other concepts such as knots and braids, and generalisations of these ideas.
Hopf-Galois theory is a generalisation of classical ideas concerning symmetries amongst the roots of polynomial equations. The modern approach to these questions employs a variety of algebraic objects including groups, rings, fields, algebras, and modules. It has applications to questions arising in number theory, and connections with numerous of topics, including skew braces.
Our work in combinatorics focusses on enumerative combinatorics, with a specialisation in permutation patterns.
Enumerative combinatorics tries to answer how many ways certain objects can be formed. Some combinatorial objects include permutations, lattice paths, graphs, set partitions, etc. Once the structure of a set of objects is well understood there exist many tools that can allow you to answer questions such as how many there are either exactly or asymptotically, or uniformly sample these objects. To find this structure, researchers have tended to adopt ad-hoc “pen-and-paper" approaches. Our research aims to automate this step, which includes building software and using tools such as formal languages, automata, computational algebra, and symbolic combinatorics.
Permutation patterns is an active area of enumerative combinatorics that has meaningful connections to computer science, statistical mechanics, and algebraic geometry. We study the structure of sets of permutations defined by the avoidance of certain patterns, and write algorithms to help solve problems in this area.
Please visit the web pages of individual members of staff for more details on their research.
- Dr P Truman
- Dr Christian Bean
Current PhD Students
- Isabel Martin-Lyons (Supervisor: Paul Truman)
- Abigail Ollson (supervisor: Dr Christian Bean)
Recent PhD Students
- George Prestidge (supervisor: Dr Paul Truman)
- Stuart Taylor (supervisor: Dr Paul Truman)
- Hopf-Galois theory and module structure of algebraic integers (Dr Paul Truman)
- Skew braces, bracoids, and the Yang-Baxter equation (Dr Paul Truman)
- Combinatorial exploration (Dr Christian Bean)
- Permutation patterns (Dr Chrisitan Bean)