The course aims to bring the student to a sound knowledge of the basics and some complements of Simplectic and Poisson Geometry, as well as certain aspects of Lie theory, and to know their relevance within Mathematical Physics. The course will focus on preparing the student for the later study of modern topics such as the theory of Lie algebroids and groupids.
Course Prerequisites
Basic notions in differential geometry: differentiable forms and vector fields on differentiable manifodls, pushforward and pullback along smooth maps. Basic notions in de Rham cohomology. A basic course in rational/analytical mechanics will be useful, but not necessary. In particular the Lagrangian and Hamiltonian formalisms. Note: in principle this course and "istituzioni di geometria" can be followed in parallel.
Teaching Methods
Standard lectures
Assessment Methods
The examination consists of an oral test, aimed at verifying the degree of understanding of the theoretical topics carried out in class, clarity of exposition but also the ability to apply these notions in concrete situations. For this reason, the student will be required to have a substantial understanding of all the theory presented, which can be verified either through questions on specific topics or through the proposal of problems concerning the course topics and solvable using the tools introduced during the lectures. The questions will be articulated on variable difficulty so as to establish the degree of depth in the acquisition of these skills The formulation of the grade will be achieved by considering the overall breadth and depth of learning, as well as the clarity of exposition and the skills demonstrated in problem solving.
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Texts
John M. Lee - Introduction to smooth manifolds Ana Cannas da Silva - Handbook of symplectic geometry (see also Ana Cannas da Silva - Symplectic geometry - Chapter 3 of Handbook of diff. Geo.) Marius Crainic, Rui Loja Fernandes and Ioan Marcut - Lectures on Poisson Geometry Alan Weinstein - Lectures on symplectic manifolds Jean-Louis Koszul and Yi Ming Zou - Introduction to symplectic geometry
Contents
Symplectic and presymplectic linear algebra. Symplectic linear group. Distinct subspaces (isotropic, coisotropic, Lagrangian) of a symplectic space. Linear coisotropic and presymplectic reduction. Lia algebras and applications in differential geometry. Symplectic geometry: (pre)symplectic differentiable manifolds and distinguished submanifolds, symplectomorphisms. Symplectic vector bundles. Symplectic geometry of cotangent bundles. Poisson manifolds and their normal forms. Poisson Cohomology.
Course Language
Italian
More information
Students in the categories identified by the project on innovative teaching will also have the opportunity to have online discussion on the course's topics, and by appointment at times to be agreed with the lecturer, as well as view the lecture notes.
