Yakov Eliashberg
Stanford University
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Primary Section: 11, Mathematics Membership Type:
Member
(elected 2003)
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Research Interests
Symplectic geometry serves as a geometric language of classical and quantum mechanics. I was lucky to witness the birth, and participate in the first steps of symplectic topology, a branch of symplectic geometry designed to answer qualitative problems in mechanics, such as the existence of periodic orbits. A tightly related topic of my research is contact geometry and topology, inspired by problems in geometric optics and non-holonomic mechanics. Problems in symplectic topology led me to the theory of functions of several complex variables, where I was able to find a complete topological characterization of affine complex manifolds. One of the most important techniques, the theory of pseudo-holomorphic curves, was introduced into symplectic topology by M. Gromov. Gromov-Witten theory combines Gromov's theory with a physics inspired algebraic formalism of E. Witten. Together with H. Hofer and A. Givental, I recently began to develop an enhanced version of the Gromov-Witten theory, called symplectic field theory (SFT). The SFT has already found many applications and we hope that many more are yet to come, particularly in such seemingly unrelated areas as the theory of completely integrable systems and low-dimensional topology.
