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Department of Mathematics

Close-up of algebraic equation on blackboard

Algebra

Algebra is the study of abstract objects arising from across mathematics, unifying apparently distant areas.

Algebra is the study of abstract structures which arise across all areas of mathematics. The same structure could be found, for example, in number theory, geometry, or topology, and used as a bridge between these subjects. Study of the structure as an abstract object in isolation also allows us to see properties which wouldn't be apparent in the original setting, and is a powerful mathematical tool. A main example of such a structure is a group, arising in the study of roots of polynomials in Galois theory, in topology as the fundamental group and in geometry in the form of symmetries, as well as in many other places in virtually every field of mathematics.

Algebra has long been strength in Manchester, with expertise across much of the subject. The deep connection algebra has with the rest of mathematics is represented in the group, and the weekly algebra seminar is a lively and well-attended event.

We are engaged in research projects relating to:

  • Representations of Lie algebras
  • Finite nonabelian simple groups
  • Coxeter groups
  • Profinite groups
  • Fusion systems
  • Modular representation theory of finite groups
  • Picard groups of finite dimensional algebras
  • Semigroups
  • Geometric group theory
  • Automata
  • Hopf algebras and quantum groups
  • Cohomology of groups
  • Module categories
  • Representations of algebras
  • Groups of finite Morley rank
  • Algebraic groups
  • Algebraic geometry
  • Invariant theory
  • Black box groups