Jan 28. 2025, 14:00 — 15:00

Many partial differential equations (PDEs) arising in physics or differential geometry can be formulated as Hamiltonian systems on infinite-dimensional symplectic manifolds. Frequently, these equations are invariant under diffeomorphisms or gauge transformations, and thus the primary interest lies in solutions considered modulo these symmetries. In the language of symplectic geometry, the solution space is identified with the symplectically reduced phase space.

In this minicourse, I will explore foundational and advanced aspects of symplectic geometry that differ significantly in infinite-dimensional settings. In particular, we will cover:

• The structure of infinite-dimensional symplectic and Fréchet manifolds, illustrated through examples from physics and geometry.
• Challenges in defining Poisson brackets in infinite dimensions, and proposed solutions (particularly on Lie groups).
• Interactions of linear symplectic geometry with various topologies ("symplectic functional analysis").
• Infinite-dimensional Lie groups and their symplectic actions on manifolds.
• Generalized formulations of momentum maps, necessary for handling symmetries such as diffeomorphism groups.
• The extension of the singular Marsden-Weinstein reduction to infinite-dimensional Fréchet manifolds.
• Applications of infinite-dimensional cotangent bundles as symplectic manifolds, with emphasis on their role in physical systems.