By Kenji Fukaya, Permanent Faculty Member, Simons Center for Geometry and Physics

*Since 1978, the Wolf Foundation awards the acclaimed, international Wolf Prize to outstanding scientists and artists from around the world. In 2020, the Wolf Prize in Mathematics was awarded jointly to Yakov Eliashberg (Stanford) and Simon Donaldson (SCGP). In this article, Kenji Fukaya shares his thoughts on the awardees esteemed contributions to differential geometry and topology.*

As one of the founders of global symplectic geometry, Yakov Eliashberg has been a leader in this field for the past 40 years. The field of symplectic geometry studies mechanics (Hamiltonian mechanics) from the geometric point of view. This area itself is very classical, but recently people found that non-trivial global symplectic geometry also exists. Importantly, among many different kinds of geometries, global geometry is rarely non-trivial

The existence of global symplectic geometry was established via a certain remarkable result by Eliashberg. Soon after this was determined, Mikhael Gromov introduced the method of pseudo-holomorphic curve. Since then, global symplectic geometry has been one of the most active areas of mathematics.

Eliashberg also obtained many basic results in contact geometry—an odd dimensional cousin of symplectic geometry. Together with Helmut Hofer and Alexander Givental, Eliashberg also proposed Symplectic Field Theory, which associates highly sophisticated algebraic structure to a contact manifold. This conjecture is now being established by mathematicians of a younger generation such as Bao-Honda, Pardon and Ishikawa.

Eliashberg’s recent research is on the flexibility of symplectic geometry. The existence of global symplectic geometry means that certain objects cannot exist in a symplectic world. The flexibility means that certain objects do exist in a symplectic world. It is ideal that we know exactly the borderline between a rigid world and a flexible world. For a long time this was a dream far from reality. New work by Eliashberg (together with his younger collaborators such as Murphy) shows that the world of symplectic geometry is more flexible than we thought before. So now we are closer to the dream.

Sir Simon Donaldson is one of the most accomplished researchers of geometry in the last 50 years. When he was just in his 20s, Donaldson found an astonishing result on 4-dimensional topology. Topologists had used linear differential equations for a long time in their research. Donaldson found that by using a certain non-linear differential equation—Yang-Mills equation—we can prove that 4-dimensional Euclidean space has an ‘exotic’ smooth structure. When this result appeared many topologists wondered whether this was a sporadic result or not. Donaldson then solved many fundamental open questions on differential topology of 4-manifolds by using Yang-Mills equation. Since then Gauge theory (such as Yang-Mills equation) plays the dominant role in the study of differential topology of 4-dimensional manifo

Donaldson also gave a fundamental contribution to symplectic geometry. Gromov discovered that one can use a complex analytic map from complex one-dimensional space to a symplectic manifold as a tool to study symplectic manifolds. On the other hand, complex analytic function on a symplectic manifold does not exist generically. Donaldson nevertheless showed we can still study symplectic manifolds by using ‘almost’ complex analytic function.

Donaldson’s recent important achievement focuses on the existence of Einstein metric on complex manifolds. One of the most important results in geometric analysis (by Yau) is the existence of Kähler metric with Ricci curvature 0 (that is, a solution of Einstein’s equation of gravity) under a certain easily checkable assumption. There is a similar result by Audin and Yau in the case of negative Ricci curvature. The case of positive Ricci curvature has long been a standing open question. Donaldson, together with his collaborators Sun and Chen, finally solved this problem.

*Congratulations to Yakov Eliashberg and Simon Donaldson on their ground-breaking contributions in geometry and topology.*