SFB TRR 191: Symplectic Structures in Geometry, Algebra and Dynamics

Basic data for this project

Description

Since their inception, the study of symplectic structures and the applications of symplectic techniques (as well as their odd-dimensional contact geometric counterparts) have benefited from a strong extraneous motivation. Symplectic concepts have been developed to solve problems in other fields that have resisted more traditional approaches, or they have been used to provide alternative and often conceptionally simpler or unifying arguments for known results. Outstanding examples are property P for knots, Cerf's theorem on diffeomorphisms of the 3-sphere, and the theorem of Lyusternik-Fet on periodic geodesics. The CRC/TRR 191 fosters the cooperation of, on the one hand, mathematicians who have been socialized in symplectic geometry and, on the other, scientists working in areas that have proved important for the cross-fertilization of ideas with symplectic geometry, notably dynamics and algebra. In addition, the CRC explores connections with fields where, so far, the potential of the symplectic viewpoint has not been fully realized or, conversely, which can contribute new methodology to the study of symplectic questions (e.g. optimization, stochastics, visualization). The CRC bundles symplectic expertise that will allow us to make substantive progress on some of the driving conjectures in the field, such as the Weinstein conjecture on the existence of periodic Reeb orbits, or the Viterbo conjecture on a volume bound for the symplectic capacity of compact convex domains in R2n. The latter can be formulated as a problem in systolic geometry and is related to the Mahler conjecture in convex geometry. The focus on symplectic structures and techniques will provide coherence to what is in effect a group of mathematicians with a wide spectrum of interests.