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Mikhail Gromov is a mathematician whose work connects Riemannian geometry, symplectic geometry and geometric group theory. His 1985 paper introduced powerful methods based on J-holomorphic curves and established the ball-embedding obstruction underlying the Gromov non-squeezing theorem.^[ref]

The obstruction says that a standard symplectic ball of radius $R$ cannot fit symplectically into $B^2(r)\times {\mathbb{R}}^{2n-2}$ when $R>r$. In higher dimensions the target has infinite volume, so preserving volume is weaker than preserving the symplectic form. This gives a concrete example of the global rigidity his work uncovered. His IHES profile records his permanent professorship from 1982 to 2015 and subsequent emeritus status.^[ref] He received the 2009 Abel Prize.^[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]