Mathematics

Smooth manifolds, maps between them, tangent vectors, submersions, immersions, and embeddings, coverings of smooth manifolds, submanifolds. Sard's and Whitney's theorems. Lie groups, vector fields. Cotangent bundle, differential forms, an introduction to symplectic manifolds.

Basic topology of R^n Basic linear algebra Differentiable multivariable calculus

Lee, John M. "Introduction to smooth manifolds". Second edition. Graduate Texts in Mathematics, 218. Springer, New York, 2013. xvi+708 pp. ISBN: 978-1-4419-9981-8. Abate, Marco, e Tovena, Francesca "Geometria differenziale". Unitext, 54. La Matematica per il 3+2. Springer, Milan, 2011. xiv+465 pp. ISBN: 978-88-470-1919-5; 978-88-470-1920-1. Lee, Jeffrey M. "Manifolds and differential geometry". Graduate Studies in Mathematics, 107. American Mathematical Society, Providence, RI, 2009. xiv+671 pp. ISBN: 978-0-8218-4815-9.

In-person lectures in class. In-person recitations in class.

Optional but strongly recommended. In person.

The written exam is held in person. Registration for exam sessions closes strictly ten days before the exam date. Those registered for the various exam sessions will receive an email in due time at their institutional email address with the information for the convocation (classroom, times, etc.). During the written test, it is essential to have a valid identity document (with a photograph), which will be checked during the exam. You can bring with you: pen(s), possibly water and snacks, notes taken during the course, as well as exercise sheets. It is not permitted to bring anything else. Students who obtain a grade higher than or equal to 18 in the written exam must take a brief interview and, optionally, complete a more substantial oral exam (essential, for example, to achieve honors). Students who receive a grade lower than 18 in the written exam will not be able to access the oral exam and must therefore register for a subsequent exam session.

In presence