BACH Seminar

Abstract: By a result due to Ziltener, there exist no closed embedded Bohr-Sommerfeld Lagrangians inside CP^n for the prequantisation bundle whose total space is the standard contact sphere. On the other hand, any embedded monotone Lagrangian torus has a canonical nontrivial cover which is a Bohr-Sommerfeld immersion. We draw the front projections for the corresponding Legendrian lifts inside a contact Darboux ball of the threefold covers of both the two-dimensional Clifford and Chekanov tori (the former is the Legendrian link of the Harvey-Lawson special Lagrangian cone), and compute the associated Chekanov-Eliashberg algebras. Although these Legendrians are not loose, we show that they both admit exact Lagrangian cobordisms to the loose Legendrian sphere; they hence admit exact Lagrangian caps in the symplectisation, which are non-regular Lagrangian cobordisms. In addition, we will discuss the conjectural relation between the superpotential of an embedded monotone Lagrangian two-torus in CP2 with the augmentation polynomial of the Legendrian lift of its canonical threefold Bohr-Sommerfeld cover. This is joint work with Georgios Dimitroglou Rizell.

Abstract: In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics via the geodesic flow of the right-invariant energy metric on the group of volume-preserving diffeomorphisms of the flow domain. We describe several recent ramifications of this approach related to compressible fluids, optimal mass transport, as well as Newton's equations on diffeomorphism groups and smooth probability densities. It turns out that various important PDEs of hydrodynamical origin can be described in this geometric framework in a natural way. This is a joint work with G.Misiolek and K.Modin.