IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar
Three 20 Minute Research Talks
Levin Maier (Heidelberg University) : From Geometric Hydrodynamics to Periodic Geodesics on Manifolds of Mappings
In this talk, we begin by recalling Arnold’s geometric formulation of hydrodynamics and then extend this framework to a broader class of Hamiltonian systems, incorporating various PDEs arising in mathematical physics. This motivates the study of infinite-dimensional manifolds and, in particular, half Lie groups: topological groups in which right multiplication is smooth while left multiplication is only continuous. Important examples include groups of $H^s$ - or $C^k$-diffeomorphisms of compact manifolds.
Within this setting, we establish several Hopf–Rinow type theorems for right-invariant magnetic systems and for certain Lagrangian systems on half Lie groups, thereby extending recent results of Bauer–Harms–Michor from the case of geodesic flows to this more general context. Finally, we show that any non-aspherical half Lie group equipped with a strong Riemannian metric necessarily admits a contractible periodic geodesic.
This talk is based partially on joint work with M. Bauer and F. Ruscelli.
Elad Kosloff (Hebrew University of Jerusalem) : Open Gromov-Witten Invariants for Even-Dimensional Lagrangians
I'll introduce the genus zero open Gromov-Witten invariants for even-dimensional Lagrangians. The definition relies on a canonical family of bounding cochains satisfying the point-like condition of Solomon-Tukachinsky, with non-commutative coefficients. In dimension two, these recover Welschinger's invariants. I'll also present computations for even dimensional quadric hypersurfaces, demonstrating that these invariants can be non-vanishing in high dimensions with multiple boundary constraints.
Joel Schmitz (Université de Neuchâtel) : Tropical Wave-Fronts & Nodal Tangles
Given a symplectic 4-manifold it may admit multiple toric fibrations. These can be seen as boundary points of the moduli space of almost toric fibrations. We will sketch that all toric fibrations are in the same connected component of this moduli space, as suggested by Symington. For this we use some results from tropical geometry.