Research Interests

My work concerns the topology of manifolds. Thirty years ago it was high-dimensional (>4) manifolds that were not necessarily differentiable or even triangulable. But I switched to work on low dimensional manifolds and in particular 4-dimensional manifolds. At first the principal issue was the classification of simply connected 4-manifolds, in both the topological and differentiable categories. Beginning in the 1970s theoretical physics began having a large impact on this subject through gauge theory and topological quantum field theory. Gauge theory provided invariants in dimension 4 that led to great progress in classification, and now we work on many questions motivated, even if only distantly, by physics. Symplectic structures in even dimensions and contact structures in dimension 3 are becoming steadily more important. My interests might be described most succinctly as low-dimensional bordism theory, where the bordisms may have a variety of extra structure, from differentiable to almost complex, and the techniques involve increasing amounts of differential geometry.

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Section 11: Mathematics