The algebra of multiplication arises naturally in a plethora of different contexts, such as symmetries and differential operators. The abstract study of these has led to the notions of groups and rings which are the foundation of modern algebra. The research interests of the Algebra group spans a wide spectrum including homological algebra, group theory, quantum groups, representation theory and noncommutative algebra.
Interested primarily on operator algebras, a cross-field of many areas of mathematics and physics, in particular von Neumann algebras, Jones' Subfactor Theory and their interactions with group theory, ergodic theory, representation theory and conformal field theory.
Interested in various noncommutative algebras arising from noncommutative algebraic geometry. These include orders, Sklyanin algebras, Clifford algebras and twisted co-ordinate rings. He has studied noncommutative Grothendieck duality theory and the McKay correspondence.
Now semi-retired, Peter has publications in algebraic geometry (localisation at fixed points). algebraic topology (related to geometrical physics), representation theory (including the first non-trivial progress towards a key finiteness conjecture in the modular representation theory of finite groups), homological algebra and the insecurity of Japanese naval ciphers in WW2. Currently his interests are returning to modular representation theory and geometrical physics.
His interests lie in the representation theories on algebraic and quantum groups, finite groups of Lie type, finite dimensional algebras, and related topics. His recent work has concentrated mainly on the Ringel-Hall approach to quantum groups and q-Schur and generalised q-Schur algebras and their associated monomial and canonical basis theory. He is also interested in combinatorics arising from generalised symmetric groups, Kazhdan-Lusztig cells and representations of finite algebras.
Interested in fusion categories and planar algebras. He is particularly interested in the representation theory of fusion categories coming from von Neumann algebras.
Interested in Coxeter groups, complex reflection groups and their Hecke algebras.
His research lies at the intersection of algebraic geometry, algebraic topology and homological algebra. In particular, he is interested in motivic cohomology, a new cohomological theory for algebraic varieties introduced by V. Voevodsky, and in the K-theory of algebraic varieties.
His interest in the theory of hypergroups has led him into description of various finite hypergroups. These have rather interesting algebraic properties and may be applied in the study of diophantine equations.