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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.