Recently strong links between mathematical physics, symplectic geometry and algebraic geometry have been established through the discovery of 'quantum cohomology'. The proposed project would be to study connections between the topology of symplectic automorphism groups and the algebraic structure of the quantum cohomology ring. This includes the development of algebro-geometric methods to make computations in Floer homology. A second part would treat analogues of the classical K-theoretical surgery obstructions (leading for example to a 'Floer simple homotopy type'). The methods to be used come from nonlinear analysis, topology and algebraic geometry. Research would be supervised by Prof. Donaldson, leading to a D. Phil thesis.