Foundations of Symplectic Field Theory

• Applicant: Professor Dr. Kai Cieliebak
• Subject Area: Mathematics
• Term: from 2009 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 157897074

• Final Report Year: 2016

Final Report Abstract

The goal of this project was to provide analytical foundations for symplectic field theory (SFT), a unified theory of holomorphic curves in symplectic manifolds, and apply them to questions in symplectic topology. The realization early in the project that the envisioned techniques are not suitable for the most general cases of SFT led to a course correction of the corresponding subproject. After this, the following results were obtained: A rigorous definition of higher genus Gromov-Witten invariants on general symplectic manifolds; • foundations for the algebraic structures arising in SFT, higher genus Lagrangian Floer homology, and string topology; • transversality for holomorphic curves in the complement of a Lagrangian embedding with applications, among them the proof of Audin’s conjecture that every Lagrangian torus in Cn has minimal Maslov number 2; • a proof of the isomorphism between the cord algebra of a knot in R3 and the Legendrian contact homology of its conormal bundle; • a partial Giroux correspondence between stable Hamiltonian structures and open books; • a new connection between the Euler equations of hydrodynamics and stable Hamiltonian structures; • a partial classification result for monotone Lagrangian tori in S2 × S2.

Publications

• Geometric transversality in higher genus Gromov-Witten theory
  A. Gerstenberger
  
• Punctured holomorphic curves and Lagrangian embeddings
  K. Cieliebak, K. Mohnke
  
• Stable Hamiltonian structures in dimension 3 are supported by open books, J. Topology 7, no. 3, 727–770 (2014)
  K. Cieliebak, E. Volkov
  (See online at https://doi.org/10.1112/jtopol/jtt044)
• First steps in stable Hamiltonian topology, J. Eur. Math. Soc. (JEMS) 17, no. 2, 321–404 (2015)
  K. Cieliebak, E. Volkov
  (See online at https://doi.org/10.4171/JEMS/505)
• Hamiltonian unknottedness of certain monotone Lagrangian tori in S2 × S2
  K. Cieliebak, M. Schwingenheuer
  
• Homological algebra related to surfaces with boundary
  K. Cieliebak, K. Fukaya, J. Latschev
  
• Knot contact homology, string topology, and the cord algebra
  K. Cieliebak, T. Ekholm, J. Latschev, L. Ng
  
• A note on the stationary Euler equations of hydrodynamics, Ergodic Theory and Dynamical Systems, 37, Issue 2, April 2017 , pp. 454-480
  K. Cieliebak, E. Volkov
  (See online at https://doi.org/10.1017/etds.2015.50)