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TOPICS IN ADVANCED GEOMETRY A

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2021/2022
Year: 
1
Academic year in which the course will be held: 
2021/2022
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Calculus, in one and several real variables, will be used from the very beginning of the lecture. A thorough knowledge of topology, as acquired in Geometry 1, is also expected. The content of the lecture Geometry 2 may as well be of help.

Final Examination: 
Orale

Verification of learning will consist of two parts:

1) A 2 hours written exam where the student is asked to solve some of the exercises assigned during the course. These exercises have a sufficiently high degree of complexity to check that the students have acquired both the ability to study the main properties of the objects introduced during the lecture and to make autonomous reasoning towards the deduction of more abstract properties;

2) A traditional oral examination, during which the students will have to show that they have acquired the basic notions and the proofs of the main theorems.

Assessment: 
Voto Finale

The purpose of this lecture is to provide the student with the basic tools of differential geometry and geometric structures. Smooth (differentiable) manifolds are the natural spaces on which one can introduce the notion of differentiability of a map, and where tools from Analysis can be extended and developed. Further, manifolds are the spaces on which geometry can be done. One can, for example, adapt ideas from plane Euclidean geometry to curved spaces, as well introduce new geometries; in many cases, it is Physics that suggests which of these geometries may play a significant role.
At the end of the course we expect that:

1) students have acquired the main notions and the fundamental theorems of the theory of differentiable manifolds, and of geometric structures defined on them;

2) based on the proofs discussed during the lectures, students are able to carry out independent reasonings of medium complexity, leading them to deduce abstract properties of the above mentioned objects;

3) students are able to investigate the main properties of the objects alluded to above in concrete situations.

The main topics of the lecture are:

1) Smooth manifolds
2) Tensors and differential forms
3) Integration of manifolds
4) Riemannian geometry: curvature and geodesics
5) Symplectic geometry: Darboux theorem and Hamiltonian mechanics

Textbooks:

1) A. Cannas da Silva. Lectures on Symplectic Geometry. Springer.
2) M. P. do Carmo. Riemannian Geometry. Birkhäuser.
3) F. W. Warner. Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, 94. Springer.

Further teaching support

1) Lecture notes written by the instructor

2) Exercises of medium-high complexity to be solved at home.

The teaching method will consist of frontal lectures, in presence and streamed. Exercises both of abstract nature and on concrete examples will be assigned regularly

For office hours, please contact the instructor at his email address, giovanni.bazzoni@uninsubria.it

Professors

BAZZONI GIOVANNI

Borrowers