# Pure Mathematics Research Group (PM)

The Pure Mathematics Group in Leicester is a cohesive group of researchers working in the areas and at the intersection of modern algebra and geometry.

## Algebra & Geometry

This group has a strong research record in modern algebra and geometry with particular emphasis on algebraic topology, homological algebra, representation theory and their interaction with non-commutative geometry, algebraic and differential geometry.

Algebraic topology was originally born in the 1920's and 30's to study various topological and geometric issues, but since then has had spectacular effect in areas such as number theory, mathematical physics, algebraic geometry, dynamics, logic, algebra, computer science and even social theory, and it continues to provide foundational insight into core areas of geometry and topology.

Homological algebra is a relatively new subject which is just over sixty years of age; however it has by now become classical. It provides a common language for experts in topology, algebraic geometry, category theory and various branches of algebra. It is also fundamental for theoretical physics, particularly those aspects related with string theory and deformation quantization. In Leicester we have a very active research group that covers both fundamental work within homological algebra and algebraic topology as well as some of its important applications to other parts of mathematics.

Representation theory is a much older subject which contains some of the most beautiful and symmetric constructions in the whole of mathematics. The representation theory of various algebraic objects can be viewed as a natural extension of their structure theory, and has innumerable applications, including within quantum mechanics, crystallography and physical chemistry. Representations of group algebras, Lie algebras, finite-dimensional associative algebras and quivers are all studied within this group, together with their connections with algebraic geometry, differential geometry and theoretical physics.

One of the exciting new areas in mathematics, which was created some 30 years ago, is non-commutative geometry. The idea here is to study non-commutative rings as if they were rings of functions on some imaginary 'non-commutative spaces'. This simple idea has turned out to be very fruitful and led to numerous advances in algebra and neighbouring fields. One of the especially useful spin-offs is the theory of quantum groups.

Minimal surfaces is an area at the intersections of many branches of mathematics. Natural examples of minimal surfaces include soap films and minimal surface theory finds applications in diverse areas of science such as chemistry and engineering. The group is hosting an international research network on ‘Visualisation, minimal surfaces and integrable systems. Further current research of members of the group includes quantum groups, stability conditions in algebra, and geometric models in representation theory, moduli spaces and algebraic stacks.

## PhD Applications

The Pure group welcomes students to study for PhD degrees in any of its research areas.