OVERVIEW

Lectures are held in Italian or English, at the students' choice. The course is addressed to students in mathematics. but it can also be also attended by students in physics.

AIMS AND CONTENT

LEARNING OUTCOMES

Basic introduction to the concepts and methods of modern differential geometry.

AIMS AND LEARNING OUTCOMES

The fundamental notions discussed in the course are those of smooth structure, tangent and cotangent bundle, vector field and tensor field, algebra of differential forms, and de Rham cohomology. Connections between the topics covered and mathematical analysis, topology, algebra, mathematical physics, algebraic topology and algebraic geometry are highlighted.

TEACHING METHODS

The course follows a traditional approach.

SYLLABUS/CONTENT

1. Smooth atlases; smooth structures; topological issues

2. Smooth manifolds

3. The inverse mapping theorem

4. Partitions of unity

5. Quotient manifolds

6. Sheaf of smooth functions

7. Cotangent space and tangent space; differential of a smooth map

8. Tangent bundle and vector fields

9. Flow of a vector field; Lie derivative

10. Multilinear algebra (tensor product of R-Modules; exterior algebra)

11. Tensor fields

12. Differential forms and Cartan differential

13. de Rham cohomology

14. Orientability; integration of differential forms and Stokes' theorem.

RECOMMENDED READING/BIBLIOGRAPHY

Detailed notes will be made available to students on the Aulaweb site.

TEACHERS AND EXAM BOARD

Exam Board

CLAUDIO BARTOCCI (President)

MATTEO PENEGINI

LESSONS

LESSONS START

According to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral exam (2 questions).

Exam schedule