Unit DIFFERENTIAL GEOMETRY
- Course
- Mathematics
- Study-unit Code
- 55A00091
- Curriculum
- In all curricula
- Teacher
- Nicola Ciccoli
- Teachers
-
- Nicola Ciccoli
- Hours
- 63 ore - Nicola Ciccoli
- CFU
- 9
- Course Regulation
- Coorte 2022
- Offered
- 2022/23
- Learning activities
- Caratterizzante
- Area
- Formazione teorica avanzata
- Academic discipline
- MAT/03
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
The course may be given in English upon unanimous request of participants. - Contents
- Abstract Manifold Theory
- Reference texts
- M. ABATE, F. TOVENA, Geometria Differenziale, Springer.
F. D'ANDREA Geometria Differenziale, Liguori
J.M.LEE, Manifolds and Differential Geometry, American Mathematical Society
Gadea/Muñoz - "Analysis & Algebra on Differentiable Manifolds: A Workbook for Students and Teachers".
Notes covering some (but not all) of the program will be provided on Unistudium.
Students with special needs are required to contact me for further informations. - Educational objectives
- This course is intended to introduce students to arguments and techniques of abstract differential geometry,
The knowledge and understanding that one is supposed to get are relative to the main techniques of calculus on abstract differentiable manifolds (not neccessarily immersed, general dimension). - Prerequisites
- Differentiable calculus in several variables is the basic tool that will be used throughout the course. A working knowledge of linear algebra and general topology is required.
- Teaching methods
- The content of the course will be fully covered in lectures. There will be exercises to learn to apply the general theoretical framework in specific examples. In general students will be asked to be an active force of their learning environment, trying to avoid sterile mnemonic and standardized calculus procedures. They will be required a positive attitude towards the teaching texts and more generally towards learning. In particular incomplete proofs, exercises of which oarts are left to the student and indications of side arguments not necessarily required for the exam are not casually present and are to be though as an intentional procedure to foster personal investigation and to stimulate a curious approach to the topic.
- Other information
- Following lectures is not compulsory but strongly advised. Office hours will be communicated at the start of the course. It will be possible to fix online office hours contacting me via e-mail.
- Learning verification modality
- The exam is an oral one of length not exceeding 120 minutes.
The following ability will be tested thorugh this exam:
the ability to apply the main notions of differential geometry in explicit examples. For this reason during the exam the student can be required to solve some short exercises.
the comprehension of theoretical issues treated in the course, their relations with other fields of math and the ability to make connections between seemingly unrelated arguments in the program. For this reason students are required to be able to proof, in a sufficiently self contained manner, the main theorems seen in the course.
the quality of exposition, the level of sharpness in the use of mathematical language, the capability of filling up autonomously details, if so required.
There will be no midterm exam. For interested students it may be possible to cover part of the programme by working on a specific project to be approved. - Extended program
- Differentiable manifolds: local charts, atlases. Differentiable maps between manifolds. Tangent and cotangent spaces. Differential of a map. Tangent and cotangent bundle: hints on the general theory of bundles. Lie groups and differentiable actions. Vector fields and differential forms. Tensor fields on manifolds. Abstract calculus on manifolds. Integration and De Rham cohomology. Time permitting: Riemannian metrics.