From the point of view of classical mechanics, the group of Hamiltonian diffeomorphisms is the set of admissible motions on a given phase space. In that context, it is natural to ask what is the least amount of energy necessary to generate such a given motion. In 1990, Hofer formalized this notion through tools from symplectic topology, leading to his eponymous metric.

Objective

The purpose of this seminar is to introduce the participants to the field of symplectic topology and the questions it asks, specifically those related to Hamiltonian diffeomorphisms. They will be able to see how the geometry of the (infinite dimensional Lie) group of Hamiltonian diffeomorphisms compares and contrasts to the finite dimensional case, and how that is related to classical mechanics; all this whilst getting to learn the less technical tools of symplectic topology.

Content

The seminar—seperated in 12 instances—will start by covering general facts on symplectic manifolds and will then move on to more and more specific statements on Hofer's geometry. Eventually, this will all circle back to some dynamical applications and will give a taste of modern symplectic topology.

Literature

L. Polterovich: The geometry of the group of symplectic diffeomorphisms. D. McDuff & D. Salamon: Introduction to symplectic topology

Prerequisites / Notice

The participant should know all the basic facts of differential geometry, specifically forms, their integration, and their differentiation. However, no prior knowledge on symplectic topology or dynamics will be assumed.

Competencies

Subject-specific Competencies

Concepts and Theories

assessed

Techniques and Technologies

assessed

Method-specific Competencies

Analytical Competencies

fostered

Problem-solving

fostered

Social Competencies

Communication

assessed

Personal Competencies

Adaptability and Flexibility

fostered

Creative Thinking

fostered

Critical Thinking

fostered

Integrity and Work Ethics

fostered

Self-awareness and Self-reflection

fostered

Self-direction and Self-management

fostered

Performance assessment

Performance assessment information (valid until the course unit is held again)