Geometry and Algebraic Topology play major roles throughout Mathematics and its applications, with geometric and topological ideas often being indispensable.

Algebraic Topology has developed important machinery such as cohomology theories including ordinary cohomology, K-theory, cobordism and elliptic cohomology.
These are often of use in geometric situations, as well as within Algebraic
Topology; Algebraic Topologists  tend to study much less `rigid' geometric
situations than other Geometers.There have also been significant interactions with many areas of Algebra, and indeed much of Algebraic Topology can be viewed as `applied algebra' as well as being a major source of innovative algebraic ideas. In the words of Hermann Weyl: `In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics'.

Departmental research activity in Topology occurs in the following areas.

• Dr A.J. Baker works on stable homotopy theory, bordism theory, the topology of classifying spaces and Thom spectra and the structure of periodic cohomology theories, in particular applying formal group theory and number theory. Some of his recent work has been on elliptic cohomology, which involves ideas from both classical and modern theories of elliptic curves and modular forms and has relations with physics as well as many areas of mathematics. Most recently, he has investigated `brave new rings’ (also known as S-algebras) and their modules, particularly questions relating to existence of such structures and their Galois theory. He also maintains the British Topology Home Page.
  
  His publications can be found here.
  
  

• Cenap Özel who obtained his PhD in 1998 with a thesis on On the Complex Cobordism of Flag Varieties Associated to Loop Groups, and they have written a joint paper.
• Mark Brightwell who obtained his PhD in 1999 with a thesis on Lattices and automorphisms of compact complex manifolds, in which he constructed manifolds with interesting finite group actions.
• Mohammed Alshumrani who obtained his PhD in 2006 with a thesis on Homotopy Theory in Algebraic Derived Categories.
• Helen Gilmour who obtained her PhD in 2006 with a thesis on Nuclear and minimal atomic S-algebras

His current students are:
Philipp Reinhard.

• Prof. P. Kropholler works on cohomology and geometry in group theory, group actions on cell complexes, analytic methods in group cohomology with applications to K-theory and the zero divisor conjecture for group algebras, complete cohomology (a generalization to infinite groups of Tate cohomology) and cohomological finiteness conditions, invariant theory (especially over finite fields and applications in algebraic topology).
  
  His publications can be found here.
  
  
• Prof. S.J. Pride uses methods of low dimensional topology to study groups given by presentations. He has also adopted some of these techniques to deal with general string rewriting systems; this work has some links with language theory and theoretical computer science.
  
  His publications can be found here.
  
  
• Dr R.J. Steiner works on branches of higher category theory related to algebraic topology.
  
  His publications can be found here.

  
  He has supervised one research student, Hongbin Cui, who obtained a PhD in 2001 for his dissertation `The omega-categories associated with products of infinite-dimensional globes'.

Seminars

During term-time there are regular meetings of the Geometry and Topology Seminar.