In 2013, he then moved to the United States in order to join the faculty of the Simons Center for Geometry and Physics at Stony Brook.

category of whose objects are Lagrangian submanifolds of a given symplectic manifold, is named after him, and is intimately related to Floer homology.

Other contributions to symplectic geometry include his proof (with Kaoru Ono) of a weak version of the Arnold conjecture and a construction of general Gromov-Witten invariants.

His many other mathematical contributions include important theorems in Riemannian geometry and work on physics-related topics such as gauge theory and mirror symmetry.

Fukaya was an invited speaker at the 1990 International Congress of Mathematicians in Kyoto, where he gave a talk entitled, Collapsing Riemannian Manifolds and its Applications.