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Symplectic topology and its interactions: from dynamics to quantization

Objective

"The proposed research belongs to symplectic topology, a rapidly developing
field of mathematics which originally appeared as a geometric tool for problems of classical mechanics. Since the 1980ies, new powerful methods such as theory of pseudo-holomorphic curves, Morse-Floer theory on loop spaces, symplectic field theory and mirror symmetry changed the face of the field and put it at the crossroads of several mathematical disciplines. In this proposal I develop function theory on symplectic manifolds, a recently emerged subject providing new tools and an alternative intuition in the field. With these tools, I explore footprints of symplectic rigidity in quantum mechanics, a brand new playground for applications of ``hard"" symplectic methods. This enterprise should bring novel insights into both fields. Other proposed applications of function theory on symplectic manifolds include Hamiltonian dynamics and Lagrangian knots. Function theory on symplectic manifolds is fruitfully interacting with geometry and algebra of groups of symplectic and contact transformations, which form another objective of this proposal. I focus on distortion of cyclic subgroups, quasi-morphisms and restrictions on finitely generated subgroups including the symplectic and contact versions of the Zimmer program. In the contact case, this subject is making nowadays its very first steps and is essentially unexplored. The progress in this direction will shed new light on the structure of these transformation groups playing a fundamental role in geometry, topology and dynamics."

Field of science

  • /natural sciences/mathematics/pure mathematics/topology
  • /natural sciences/mathematics/pure mathematics/geometry
  • /natural sciences/physical sciences/quantum physics
  • /natural sciences/physical sciences/classical mechanics

Call for proposal

ERC-2013-ADG
See other projects for this call

Funding Scheme

ERC-AG - ERC Advanced Grant

Host institution

TEL AVIV UNIVERSITY
Address
Ramat Aviv
69978 Tel Aviv
Israel
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 787 200
Principal investigator
Leonid Polterovich (Prof.)
Administrative Contact
Lea Pais (Ms.)

Beneficiaries (1)

TEL AVIV UNIVERSITY
Israel
EU contribution
€ 1 787 200
Address
Ramat Aviv
69978 Tel Aviv
Activity type
Higher or Secondary Education Establishments
Principal investigator
Leonid Polterovich (Prof.)
Administrative Contact
Lea Pais (Ms.)