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The ANTLR group has broad interests, including representation theory, model theory, number theory, set theory, decidability problems, and formalization.
Algebra
Algebra is the study of structures which appear in different parts of mathematics, often originating in physics. Starting with semigroups which study composable operations, we can add requirements (such as the operations being reversible, which yields so-called groups) or more structures with various conditions on their interactions, yielding rings, fields, algebras, Hopf algebras, categories, etc.
Number Theory
Number Theory is the study of the integers and their properties, including understanding prime numbers and integer solutions to equations. To do so, we associate various algebraic or geometric structures and study their behaviour, allowing us to draw conclusions about the original problem.
Logic
Logic encompasses a variety of topics at the foundations of mathematics. Model theory studies formal theories and structures which reflect these theories. Set theory is the basis for most modern mathematics and studies axioms about the infinite. Decision problems study whether there is an algorithm which will answer a mathematical question: for example, whether two words represent the same element of a group or semigroup.
Representation theory
Representation theory is a way of studying algebraic structures by investigating how they can act on other structures. The most basic example is studying abstract symmetries by their action on a physical object. More generally, we can study a given algebraic structure by its actions on one that can be more easily understood.
There are many connections among our research in these broad areas, as well as beyond. For example, we do research: on the formalization of mathematical proof in LEAN, with a particular focus on number theoretic applications; on the model theory of representations; on the application of methods from combinatorial algebra, geometric group theory, algebraic topology, and formal language theory to study decision problems; on the applications of representation theory to number theory via the local Langlands program; on applications of model theory to number theory and diophantine geometry; on connections between combinatorics and representation theory; on category-theoretic methods in representation theory via categorification; on category-theoretic methods in model theory.
As well as our faculty members, we have many postgraduate research students and several postdoctoral researchers. We have a weekly ANTLR seminar for external speakers, as well as more informal specialised seminars and usually one or two study groups at any one time, and we regularly organise workshops and conferences, including SEEMOD (South and East of England Model Theory).
To read more about our research, please see individuals’ webpages, which you can find through the list of group members below.
Group Head: Shaun Stevens
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