On May 27 the 2025 Shaw Prize was announced in Hong Kong. Kenji Fukaya, Professor at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) and at the Yau Mathematical Sciences Center of Tsinghua University, was awarded the Shaw Prize in Mathematical Sciences for his revolutionary work in symplectic geometry, above all for foreseeing and then building the category now named after him—the Fukaya category—whose objects are Lagrangian submanifolds in a symplectic manifold. He led the arduous campaign to construct this category and went on to break new ground in symplectic topology, mirror symmetry, and gauge theory.

Professor Hélène Esnault, chair of the Shaw Prize Mathematical Sciences Selection Committee, told the press conference that in classical mechanics the time evolution of a physical system is described by a flow on phase space determined by a Hamiltonian function. In the 1960s Vladimir Arnold framed a series of conjectures about lower bounds for the number of periodic solutions of this flow when the Hamiltonian is time-periodic. In modern mathematics, phase space is generalized to a symplectic manifold, and a refined conjecture gives a lower bound for the number of intersection points of two Lagrangian submanifolds.

In the 1980s Andreas Floer, inspired by infinite-dimensional Morse theory, created Lagrangian Floer theory as a route to Arnold’s conjecture. Under suitable assumptions on the symplectic and Lagrangian data, he built Floer homology from the solution space of a nonlinear partial differential equation—viewed as a moduli space—and used it to settle several special cases of Arnold’s conjecture. Without those assumptions, however, the moduli spaces can be highly singular and complicated, and the general case remained elusive.

Fukaya, together with many collaborators, established and vastly extended Lagrangian Floer theory—one of his principal achievements. Around 1993, drawing on Morse homotopy, he uncovered a higher-order algebraic structure in these intricate moduli spaces and proposed the bold vision of endowing every symplectic manifold with an A-∞ category—the now-famous Fukaya category. At the time, most of the tools needed to realize this vision were missing. A central difficulty was how to handle the singularities of the moduli spaces. Fukaya introduced and developed the theory of Kuranishi structures, and with several co-workers built a new method that equips singular spaces carrying such structures with virtual fundamental chains and an intersection theory for them. Together they overcame enormous challenges, and their achievement stands as a milestone in mathematics.

The Fukaya category is not only intrinsically beautiful but also an extremely powerful tool in symplectic topology. Fukaya and his collaborators have obtained striking new results on the non-displaceability of certain Lagrangian submanifolds and have constructed new quasi-isomorphisms on the Hamiltonian diffeomorphism groups of some symplectic manifolds.

One reason the Fukaya category has attracted attention across fields is Maxim Kontsevich’s homological mirror symmetry conjecture, which asserts an equivalence between the Fukaya category of a Calabi–Yau manifold and the derived category of coherent sheaves on its mirror. Fukaya has made revolutionary contributions to mirror symmetry, especially through his introduction of Floer homology.

Fukaya’s early work produced major advances in Riemannian geometry and gauge theory. More recently, he and his collaborators have used Lagrangian Floer theory to achieve a breakthrough on the Atiyah–Floer conjecture for three-manifolds—one of the very questions that motivated Fukaya to invent his category.

Kenji Fukaya received his B.S. in mathematics from the University of Tokyo in 1981 and his Ph.D. there in 1986 under Akio Hattori. From 1983 to 1993 he held research and associate positions at the University of Tokyo, moved to Kyoto University as professor in 1994, and became a permanent member of the Simons Center for Geometry and Physics at Stony Brook University in 2013. In the fall of 2024 he took up full-time professorial appointments at both the Yau Mathematical Sciences Center of Tsinghua University and the Beijing Institute of Mathematical Sciences and Applications. Among the many honors he has received are the 1989 Geometry Prize of the Mathematical Society of Japan, the 2002 Inoue Prize for Science, the 2003 Japan Academy Prize, the 2009 Asahi Prize, and the 2012 Fujiwara Prize. Shing-Tung Yau has remarked that Heisuke Hironaka, Shigefumi Mori, Kenji Fukaya, and Masaki Kashiwara are the four Japanese mathematicians whose work has astonished the world; Fukaya’s contributions to Riemannian geometry in his early career and to symplectic geometry since the 1990s have provided essential tools for low-dimensional topology and mirror symmetry and have opened and guided entire frontiers of modern geometry.

Established in accordance with the wishes of Sir Run Run Shaw, the Shaw Prize is awarded annually to scientists who have achieved recent breakthroughs whose impact on human life is already profound. The inaugural ceremony took place in Hong Kong in 2004. Over twenty-one editions, the Mathematical Sciences award has honored thirty-two mathematicians, beginning with Shiing-Shen Chern and including Fields medalists Andrew Wiles, David Mumford, Maxim Kontsevich, and, in 2023, Shing-Tung Yau himself. The Shaw Prize website observes that mathematics is the universal language of all natural sciences and modern technologies, that its twentieth-century advances have opened new worlds and solved old impossibilities, and that its influence now permeates every creative science and technology as well as the social sciences; with the rise of computer science, information technology, and statistics, mathematics will only become more vital to humanity in the twenty-first century.

The 2025 Shaw Prizes are given in three categories—Astronomy, Life Science & Medicine, and Mathematical Sciences. This year’s awards will be presented in Hong Kong on 21 October. John Richard Bond (Canadian Institute for Theoretical Astrophysics and University of Toronto) and George Efstathiou (University of Cambridge) share the astronomy prize, Wolfgang Baumeister (Max Planck Institute of Biochemistry, emeritus) receives the life-science-and-medicine prize, and Kenji Fukaya stands alone as the 2025 Shaw Laureate in Mathematical Sciences.