CODE 61467 2023/2024 6 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 3 MATEMATICA 8760 (L-35) - GENOVA MAT/03 Italian GENOVA 2° Semester AULAWEB

## 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.

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

## TEACHERS AND EXAM BOARD

### Exam Board

CLAUDIO BARTOCCI (President)

MATTEO PENEGINI

## LESSONS

### Class schedule

L'orario di tutti gli insegnamenti è consultabile all'indirizzo EasyAcademy.

## EXAMS

### EXAM DESCRIPTION

Oral exam (2 questions).

### Exam schedule

Data Ora Luogo Degree type Note
27/05/2024 09:00 GENOVA Esame su appuntamento