Mathematics advances by recognising both the individuality of its structures and the principles they share. A theorem about groups may reveal a pattern that also belongs to rings, Lie algebras or ordered structures; a resemblance between examples may, in turn, conceal a decisive difference. Our research seeks to understand what is unique and what is general about mathematical structures. Category theory offers a perspective on the large-scale structure of mathematics, studying objects through the relationships between them. In dialogue with algebra, this perspective allows us to ask why familiar constructions work, which assumptions they require, and how far their underlying ideas extend.

This dialogue is central to categorical algebra: the study of algebra from a broader structural point of view. Concrete examples, including groups, rings, Lie algebras, lattices and Heyting semilattices, inform the search for general principles. Universal algebra approaches such questions through operations and identities. Our work on Mal’tsev conditions and categorical exactness properties explores how these algebraic laws are reflected in the behaviour of relations and constructions. One strand develops a matrix-based classification of exactness properties, bringing Mal’tsev, majority and related categories into a common framework and investigating which properties imply others.

Questions of structure are also questions of how objects can be assembled and decomposed. Our research on extensive and coextensive categories examines the behaviour of coproducts and products, connecting categorical principles with refinement and decomposition properties in algebra. A complementary direction concerns locally algebraically cartesian closed categories and algebraic exponentiation. Work on Lie algebras and their categorical generalisations provides examples of this phenomenon, including Lie superalgebras and differential graded Lie algebras. Such investigations move between the specific features of an algebraic setting and the categorical conditions that explain them.

These interests lead naturally to connections with order, topology and logic. Research on Artin glueings of frames relates the assembly of spaces in point-free topology to algebraic ideas of extensions and semidirect products. Another line of work uses noetherian forms to study homomorphism theorems, products and commutators in a setting that makes duality explicit. Across these directions, familiar algebraic ideas become a means of discovering connections between apparently different mathematical worlds.

For students, this field offers an opportunity to combine the discipline of concrete calculation and examples with the search for conceptual explanations. A useful abstraction must account for what happens in particular cases, while a well-chosen example can expose the limits of a general theory. The continuing exchange between these two viewpoints shapes our research: understanding not only that a mathematical pattern occurs, but why it occurs, where it persists, and what makes its different manifestations distinctive.