Austrian-Polish FWF-NCN-Project I-5015-N 

2021–2025, 306.768 EUR

Wider research context :

In the last years there is a trend to apply the methods of functional analysis to the

geometry in order to create a rigorous setting for Hamiltonian mechanics on infinite

dimensional manifolds. The aim is to create a mathematically consistent framework in

which both quantum mechanics and integrable systems can be studied.

Approaches :

The research lies in the field of mathematical physics and it employs both methods

of differential geometry and functional analysis. It concerns with infinite-dimensional

counterparts of geometric structures which lie at the foundation of classical mechanics.

Hypotheses/Research/Objectives :

Hamiltonian mechanics is part of Poisson geometry. The natural action of Lie groups

on phase spaces of classical systems leads to the notion of Poisson–Lie groups. For

systems with an infinite number of degree of freedom, it is natural to study the concept

of Poisson–Lie groups in the framework Banach geometry. Structures related to the

restricted Grassmannian are key examples in the understanding of this theory.

Originality :

The theory of Banach Poisson–Lie groups that we intend to explore is a new concept in

the context of infinite-dimensional geometry. Extension to the Fréchet context will allow

to study hamiltonian systems coming from gauge theories. 

There are many interesting applications to be discovered.

Primary researchers involved :

Alice Barbora Tumpach (WPI, Vienna) 

Tomasz Goli´nski (University of Bialystok,Poland)