IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

Emilia Konrad (Augsburg University) : Construction of Constrained Floer Homology

We consider the symplectic area functional, constrained to loops of vanishing Hamiltonian mean value: It has the same critical points as the Rabinowitz action functional, and can be used to define a similar Floer homology. In contrast to RFH, it admits an intrinsic product structure, but also involves a non-local term in the gradient flow equation.
This talk will delve into the Fredholm and compactness results required to define CFH, and also discuss some remaining „mysteries“.

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.

Ciprian Bonciocat (Stanford University) : Degenerate Lagrangian Intersections and Parametric Floer Homotopy Theory

In this talk I will introduce the idea of Floer homotopy theory and show how it can be used to give lower bounds on degenerate Lagrangian intersections, in the case of plumbings of cotangent bundles along a submanifold. The strength of the invariant comes from incorporating all additional choices involved in the construction of the stable homotopy type, resulting in a parameterized spectrum. The work is joint with Kenneth Blakey.