Special Seminar

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Elliptic methods in the study of Hamiltonian systems

Speaker: Umberto Hryniewicz, RWTH Aachen University

Location: Online

Videoconference link: https://nyu.zoom.us/j/93345324049

Date: Tuesday, November 18, 2025, 11 a.m.

Synopsis:

This talk will describe how elliptic PDE methods can be used to analyze the variational structure of the action functional in Hamiltonian mechanics. It will also present some recent significant applications of these methods to the study of Hamiltonian dynamics. Firstly, we will present a Poincaré–Birkhoff type theorem for Hamiltonian flows on star-shaped 3D energy levels. This work was presented at the Geometry Session of the ICM 2018 in Rio de Janeiro. The second result to be presented is a proof of the 'two or infinity' dichotomy for periodic orbits of Reeb flows on closed, connected 3-manifolds provided that the first Chern class of the contact structure is torsion. This resolves a long-standing conjecture in differential geometry, which asserts that a Finsler metric on the 2-sphere admits either exactly two or infinitely many closed geodesics. Finally, we will discuss a generic existence result for global sections of Reeb flows on closed 3-manifolds, whose proof combines holomorphic curve methods and ergodic theory

Zoom Link:  https://nyu.zoom.us/j/93345324049