Prof John Greenlees

Position: Professor
Home page:
Telephone: (0114) 2223786
Office: J15 Hicks building
Photo of John Greenlees


MAS435 Algebraic Topology Information Home page
MAS6370 Algebraic Topology I Information Home page


Interests: Equivariant topology, commutative algebra, algebraic topology
Research groups: Topology, Algebra, Category Theory
Publications: Preprint page, ArXiv, MathSciNet


Current grants, as Principal Investigator
Rigidity and Hasse-Tate models in algebra, geometry and topology
Past grants, as Principal Investigator
Rational equivariant cohomology theories
Algebraic Rational G-Equivariant Stable Homotopy Theory for Profinite Groups and Extensions of a Torus
Orientability and Complete Intersections for Ring Spectra EPSRC
Higher Structures on Elliptic Cohomology EPSRC
Connective real K theory, compact Lie groups and geometry
Applications of local cohomology to algebraic topology
Connective K-theory of classifying spaces and geometry


Director of Research


Professor Greenlees was awarded his PhD by the University of Cambridge (1985). After a year as a Senior Rouse Ball Student at Trinity College, he spent 1986-89 at the National University of Singapore and 1989-90 at the University of Chicago. He moved to Sheffield in 1990, being awarded a Personal Chair in 1995, and has held visiting positions at the University of Chicago and the Isaac Newton Institute. He was Research Professor at MSRI (Berkeley) in 2014, and Visiting Researcher in CRM (Barcelona) and HIM (Bonn) in 2015.
Professor Greenlees has been Vice President of the London Mathematical Society since 2009, was a member of the REF2014 Mathematical Sciences Subpanel, and is an editor of five journals and a book series. In the past he served as a member of the EPSRC Strategic Advisory Team (2004-06), on the Pure Mathematics Subpanel for RAE2008. He was Head of School (2010-2013) abd the Department of Pure Mathematics (2004-08).
Professor Greenlees was awarded a Nuffield Foundation Science Research Fellowship for 1995-96, and the Junior Berwick Prize of the London Mathematical Society in 1995.

Research interests:

Algebraic topology is principally concerned with geometric and topological problems but, as its name suggests, it uses methods from various parts of algebra. Traditionally, this algebra has mostly been homological, but Professor Greenlees is concerned with the topology of spaces with symmetry, so group theory also plays a part. He has been a pioneer in the use of commutative algebra in homotopy theory.

The idea of algebraic topology is to replace a slippery geometric object by a more rigid and tractable algebraic invariant. For example, compact surfaces are classified by their Euler characteristics. Accordingly the subject is principally concerned with invariants:
  • with the problem of defining invariants,
  • with the theoretical effectiveness of invariants in solving a geometric problems,
  • with the calculation of invariants, and
  • with the algebra relevant to the invariants.
By the time the problem reaches an algebraic topologist the geometry has usually been converted to structure invariant under homotopy equivalence, so most of their time will be spent finding algebra which captures the essence of the structure and then proving the critical algebraic theorems.

Professor Greenlees is particularly concerned with the topology of spaces with a group action. The group actions and topology interact in fascinating and unexpected ways, and this leads to new insights in algebra as well as topology. The relevant algebra is relatively undeveloped and is rich enough to include representation theory and commutative algebra.