The logic group’s research includes set theory, model theory, and decidability problems.
In set theory, our main interests are forcing, forcing axioms, large cardinals, the axiom of choice, combinatorial set theory, and the connections between these areas. We also work on foundational issues and the philosophy of set theory. Recent work includes the development of forcing techniques involving side conditions, the study of connection of strong forcing axioms with other axioms, and the continuing development of iteration methods in the study of choiceless set theory, as well as the applications of these methods.
Our interests in model theory include abstract stability and classification theory, and applications of model theory to other parts of mathematics. In stability theory, we often look beyond classical first-order logic to theories in positive logic or infinitary logics. We also look at category-theoretic methods in model theory. Applications of particular interest include the model theory of the complex exponential function and related analytic functions, and to number theory, particularly functional transcendence theory and Diophantine geometry.
In algorithmic problems for groups and semigroups, our main interests are decision problems for groups, monoids and inverse semigroups, including the word problem, the subgroup membership problem, the submonoid and rational subset membership problems, and the Diophantine problem. We are especially interested in developing and applying methods from combinatorial algebra, geometric group theory, algebraic topology, and formal language theory, to study these decision problems. Recent results include both decidability and undecidability results for one-relator groups and inverse monoids, for free idempotent generated semigroups, and applications of string rewriting methods to solve algorithmic problems for Plactic monoids and algebras.
The logic group has numerous postgraduate students and several postdoctoral research associates. We have our own research seminar which meets once a week in term time and regularly organises seminars and workshops. Our members participate in the SEEMOD (South and East of England Model Theory) and STUK (Set Theory in the UK) research networks.