Mathematics Research Group

Algebra, Geometry, and Discrete Mathematics theme

Significant investment over the past two years has enabled this theme to double its size and expand its scope accordingly.

From a core base of expertise in semigroup theory, combinatorial group theory, graph theory and combinatorics, our theme now includes researchers who work in commutative algebra, algebraic geometry and finite group theory, and who research applications of finite field theory to network coding.

The theme’s research in algebra concerns group theory (Dr Litterick, Prof. Williams), semigroup theory (Prof. Higgins, Dr Vernitski), and meets algebraic geometry and geometric topology (Dr Martinez-Garcia, Dr Vernitski, Prof. Williams), graph theory and combinatorics (Dr Penman, Prof. Higgins, Dr Vernitski, Prof. Williams), computational algebra and topology (Dr Litterick, Dr Martinez-Garcia, Dr Vernitski, Prof. Williams) algebraic number theory and linear algebra (Prof. Williams), and applications such as algebraic data compression methods such as Burrows-Wheeler transformations and Bloom filters (Prof. Higgins, Dr Vernitski).

Dr Martinez-Garcia currently holds an EPSRC Standard Grant which funds a Senior Research Officer (Dr Duarte Guerreiro) to carry out work on the Calabi problem for smooth Fano threefolds. Professor Williams and Dr Vernitski have each recently held Leverhulme Research Project Grants for work on group presentations with cyclic symmetries, and machine learning in knot theory.

Through its expansion, the theme has created synergies to carry out research at the interfaces of its expertise. For example, it recently hosted a workshop on Discrete Computational Mathematics (Dr Litterick, Dr Martinez-Garcia, Prof. Williams) and, in addition to sole supervisions, members co-supervise PhD students working on problems in Geometric Invariant Theory (Dr Litterick, Dr Martinez-Garcia), finite quotients of infinite groups (Dr Litterick, Prof. Williams), and algebraic invariants of graphs (Dr Penman, Prof. Williams).

Recent papers



A thick piece of twisted brown rope tied in to a large knot.
Project: Machine learning for recognising tangled 3-D objects

This project, funded by the Leverhulme Trust, will help teach computers how to understand 3-dimensional space without using deep neural networks.