TOPICS IN ADVANCED GEOMETRY A
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.
The examination will take place on two levels:
1) The assignment of a topic, agreed between student and instructor, which will be presented in an oral examination. The goal is to check the ability of the student to develop, in an autonomous way, concepts which are similar to those presented in the theoretical lectures.
2) A traditional oral examination, during which students are required to explain the basic definitions of the course and to discuss the proofs of the main theorems. It will also be required to solve some simple exercises, similar to those assigned during the semester.
The two oral examinations can happen at different moments.
The final mark, to be expressed over 30 points, is a global assessment of each part of the oral examination.
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 students are expect to:
1) have acquired the main notions and the fundamental theorems of the theory of differentiable manifolds, and of geometric structures defined on them;
2) be able, based on the proofs discussed during the lectures, to carry out independent reasonings of medium complexity, leading them to deduce abstract properties of the above mentioned objects;
3) be 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: examples and constructions, tangent space and bundle, vector fields, and integral curves.
2) Tensors and differential forms: tensor algebra, exterior algebra, exterior differential, Lie derivative.
3) Integration of manifolds: Stokes’ theorem
4) Theory of connections: affine connections, parallelism, parallel transport, torsion, curvature, and geodesics.
5) Riemannian metrics: metric connections, Levi-Civita connection, Riemann’s curvature tensor
The teaching modalities consist of standard lectures, which could be recorded, if necessary.
The teaching method consists of:
1) theoretical lectures, during which I will provide the students with the key notions of the course.
2) Regular exercise assignments, to apply the theoretical content of the lecture. The exercises can be worked out individually, or in small groups.
3) Problem sessions, during which we will discuss the solutions of the exercises; ideally, the students will play an important part in the problem session.
For office hours, please contact the instructor at his email address, giovanni.bazzoni@uninsubria.it
Professors
Borrowers
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Degree course in: MATHEMATICS
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Degree course in: MATHEMATICS