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Denis Auroux studies symplectic geometry, with work linking geometric constructions to topology and algebraic geometry. His Harvard research profile identifies the specialization.^[ref]

His lectures with Ivan Smith develop descriptions of symplectic manifolds through Lefschetz pencils and branched covers of the projective plane. The resulting invariants combine the algebra of monodromy with geometric information from pseudoholomorphic curves. This creates a bridge between concrete maps and the classification of manifolds.^[ref]

Ivan Smith works on symplectic topology. His Cambridge profile identifies him as a geometer working in that subject.^[ref]

With Denis Auroux, he wrote lectures connecting Lefschetz pencils, branched covers and symplectic invariants. A Lefschetz pencil describes a manifold by a family of surfaces with controlled singular members; blowing up the base points produces a fibration. The approach makes a global geometric object accessible through fibres, singularities and the way the fibres fit together.^[ref]