ID:
512091
Duration (hours):
48
CFU:
6
SSD:
Fisica matematica
Year:
2026
Overview
Date/time interval
Primo Semestre (23/09/2026 - 13/01/2027)
Syllabus
Course Objectives
The course aims to bring the student to a sound knowledge of basics Simplectic and Poisson Geometry, as well as certain aspects of Lie theory, and to know their relevance within Mathematical Physics. This lays the foundations for subsequent developments within geometric mechanics and field theory, as well as modern differential geometry and ‘higher structures’.
Course Prerequisites
Basic notions in differential geometry: differentiable forms and vector fields on differentiable manifolds, 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 ‘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.
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.
Texts
John M. Lee - Introduction to smooth manifolds
Ana Cannas da Silva - Lectures on symplectic geometry (see also Ana Cannas da Silva - Symplectic geometry, overview written for the Handbook of Differential Geometry (eds. F.J.E.Dillen and L.C.A.Verstraelen), Elsevier, 2005)
Marius Crainic, Rui Loja Fernandes and Ioan Marcut - Lectures on Poisson Geometry
Jean-Louis Koszul and Yi Ming Zou - Introduction to symplectic geometry
J. Marsden, T. Ratiu, Introduction to mechanics and symmetry, Springer 1994
Mark Adler, Pierre van Moerbeke, Pol Vanhaecke (auth.) - Algebraic Integrability, Painlevé Geometry and Lie Algebras (2004, Springer)
Juan -Pablo Ortega, Tudor S. Ratiu - Momentum Maps and Hamiltonian Reduction, Progress in Mathematics
Qualsiasi altro testo di geometria differenziale, per esempio L. Tu "Differential geometry", Springer
Ana Cannas da Silva - Lectures on symplectic geometry (see also Ana Cannas da Silva - Symplectic geometry, overview written for the Handbook of Differential Geometry (eds. F.J.E.Dillen and L.C.A.Verstraelen), Elsevier, 2005)
Marius Crainic, Rui Loja Fernandes and Ioan Marcut - Lectures on Poisson Geometry
Jean-Louis Koszul and Yi Ming Zou - Introduction to symplectic geometry
J. Marsden, T. Ratiu, Introduction to mechanics and symmetry, Springer 1994
Mark Adler, Pierre van Moerbeke, Pol Vanhaecke (auth.) - Algebraic Integrability, Painlevé Geometry and Lie Algebras (2004, Springer)
Juan -Pablo Ortega, Tudor S. Ratiu - Momentum Maps and Hamiltonian Reduction, Progress in Mathematics
Qualsiasi altro testo di geometria differenziale, per esempio L. Tu "Differential geometry", Springer
Contents
Linear symplectic and presymplectic algebra. Linear coisotropic an presymplectic reduction. Symplectic geometry: smooth (pre)symplectic manifolds and distinguished submanifolds, symplectomorphisms. Symplectic vector bundles. Symplectic geometry of cotangent bundles. Basics of singular foliations, equivalence classes and quotients, Godement’s criterion. Basics of Lie theory and actions on smooth manifolds. Hamiltonian actions, momentum maps, integrals of motions and symmetry prioperties (Hamiltonian Noether theorem). Symplectic/Hamiltonian reduction and Marsden—Weinstein theorem. Poisson geometry, Schouten brackets, Weinstein splitting theorem and normal forms, integrable systems (general and according to Liouville) in Poisson manifolds, Arnold-Liouville theorem in Poisson manifolds.
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.
Degrees
Degrees
MATHEMATICS
Master’s Degree
2 years
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