# Yakov Eliashberg

Wolf Prize Laureate in Mathematics 2020

**Yakov Eliashberg**

#### Affiliation at the time of the award:

**Stanford University, USA**

#### Award citation:

**“for their contributions to differential geometry and topology.”**

#### Prize Share:

**Yakov Eliashberg**

**Simon Donaldson **

Yakov Eliashberg is one of the founders of symplectic and contact topology, a discipline originated as mathematical language for qualitative problems of classical mechanics, and having deep connections with modern physics. The emergence of symplectic and contact topology has been one of the most striking long-term advances in mathematical research over the past four decades. Eliashberg is among the main exponents of this development.

Professor Eliashberg was born in 1946 in Leningrad (now St. Petersburg), Russia. He received his doctoral degree in Leningrad University in 1972 under the direction of V.A. Rokhlin, and in the same year he joined Syktyvkar University in northern Soviet Union. Eliashberg’s route passed through the refusenik years in Leningrad (1980-1987) where he had to do software engineering in order to feed his family, and where he was virtually cut off from normal mathematical life. In 1988 he emigrated to the United States and in 1989 became a Professor at Stanford University. He is a Member of U.S. National Academy of Sciences. For his contributions, Eliashberg has received a number of prestigious awards, including the Guggenheim Fellowship in 1995, the Oswald Veblen Prize in 2001, the Heinz Hopf Prize in 2013 and the Crafoord Prize in 2016. Eliashberg is currently the Herald L. and Caroline L. Ritch Professor at Stanford University.

Eliashberg developed a highly ingenious and very visual combinatorial technique that led him to the first manifestation of symplectic rigidity: the group of symplectomorphisms is closed in the group of all diffeomorphisms in the uniform topology. This fundamental result, proved in a different way also by Gromov and called nowadays the Eliashberg-Gromov theorem, is considered as one of the wonders and cornerstones of symplectic topology. In a series of papers (1989-1992), Eliashberg introduced and explored a fundamental dichotomy “tight vs overtwisted” contact structure that shaped the face of modern contact topology. Using this dichotomy, he gave the complete classification of contact structures on

the 3-sphere (1992). In these papers Eliashberg laid foundations of modern contact topology and introduced mathematical language which is widely used by researchers in this rapidly developing field.

In a seminal 2000 paper Eliashberg (with Givental and Hofer) pioneered foundations of symplectic field theory, a powerful, rich and notoriously sophisticated algebraic structure behind Gromov’s pseudo-holomorphic curves. It had a huge impact and became one of the most central and exciting directions in symplectic and contact topology. It has led to a significant progress on numerous areas including topology of Lagrangian submanifolds and geometry and dynamics of contact transformations, and it exhibited surprising links with classical and quantum integrable systems.

In recent years (2013-2015), Eliashberg found a number of astonishing appearances of homotopy principle in symplectic and contact topology leading him to a solution of a number of outstanding open problems and leading to a “mentality shift” in the field. Before these developments the consensus among experts was that the symplectic world is governed by rigidity coming from Gromov’s theory of pseudo-holomorphic curves or, equivalently, by Morse theory on the loop spaces of symplectic manifolds. The current impression based on Eliashberg’s discoveries is that rigidity is just a drop in the ocean of flexible phenomena.

Professor Yakov Eliashberg is awarded the Wolf Prize for his foundational work on symplectic and contact topology changing the face of these fields, and for his ground-breaking contribution to homotopy principles for partial differential relations and to topological foundations of multi-dimensional complex analysis.