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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]

Robert E. Gompf studies four-dimensional topology, symplectic topology and contact topology. His University of Texas research profile also identifies Stein surfaces as a focus.^[ref]

His symplectic sum construction glues symplectic manifolds along suitable codimension-two submanifolds. Among its consequences, every finitely presented group occurs as the fundamental group of a compact symplectic manifold in each even dimension at least four. This shows how flexible symplectic geometry can be compared with the additional restrictions imposed by a Kähler manifold.^[ref]

Symplectic topology studies global properties of symplectic manifolds and maps preserving their symplectic forms. Local coordinates put every symplectic form in the same standard form, by the Symplectic Darboux theorem; global embedding and deformation problems can nevertheless have rigid obstructions.

A striking example is the Gromov non-squeezing theorem. In ${\mathbb{R}}^{2n}$ with its standard symplectic form, the ball $B^{2n}(R)$ embeds symplectically into the cylinder $B^2(r)\times {\mathbb{R}}^{2n-2}$ only if $R\le r$. For $n\ge 2$ the cylinder has infinite volume, so ordinary volume comparison cannot explain the obstruction. The selected two-dimensional plane is a conjugate coordinate plane. Tools such as J-holomorphic curves connect this global rigidity to topology and Hamiltonian mechanics.^[ref]