Symplektisk geometri och differentialtopologi
Tidsperiod: 2013-01-01 till 2016-12-31
Projektledare: Tobias Ekholm
Budget: 3 600 000 SEK
Symplectic and contact geometry has its roots in the mathematical description of mechanics. Its modern development was initiated in the early 1980´s with conjectures by Arnold and then shaped by Gromov´s theory of holomorphic curves and its morsification by Floer. In 2000 Eliashberg, Givental, and Hofer introduced Symplectic Field Theory (SFT) which is a general scheme for holomorphic curves in symplectic and contact geometry with strong connections to physical theories such as supersymmetric QFT and string theory. The research program will further advance these powerful techniques and also apply them to extract differential topological information. A recent breakthrough by the applicant and Smith gives a symplectic geometric characterization of even dimensional spheres up to diffeomorphism (in particular beyond homotopy theory) and new striking results on the conjectures of Arnold. These results use more refined properties of moduli spaces than what is customary in SFT. The program will study moduli spaces in connection with differential topology, both more refined and classical properties. In particular, the program continues the work of the applicant, Bourgeois, and Eliashberg on Weinstein manifolds, where the simplest versions of SFT were computed from Lagrangian handle attaching data, to other levels of SFT, aiming at a deeper understanding of Weinstein manifolds, and via Donaldson hypersurfaces, also closed symplectic manifolds and corresponding physical theories.