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ADVANCED GEOMETRY B

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2020/2021
Year: 
1
Academic year in which the course will be held: 
2020/2021
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Elementary differential geometry of surfaces. Some basic notions of differentiable varieties

Final Examination: 
Orale

At the end of the course, the student will give a seminar on a topic agreed with the teacher for the preparation of which the concepts and methods developed in the course will be used in an essential way.
The maximum score Of 30/30 e lode is obtained if the student, in answering questions from the teacher during his seminar, proves to have understood not only the topic of the seminar, but also the concepts and methods used to prepare it, and the related mathematical proofs.

Assessment: 
Voto Finale

TRAINING OBJECTIVES.
The course aims to provide advanced theoretical training on the theory of Riemann surfaces and their module spaces, through the point of view of Teichmuller theory.
EXPECTED LEARNING RESULTS.
The student will be able to orient himself in the recent mathematical literature on the subject in question. The acquired notions can be useful for an introduction to some current research topics in complex or algebraic geometry or theoretical physics, with particular reference to string theory.

Riemann surfaces. Uniformization theorem. Complements of sheaf theory and Riemann-Roch theorem. Plane hyperbolic geometry. Hyperbolic geometry of Riemann surfaces. Fundamental domains. Quasi-conformal maps. Extremal maps of Teichmuller. Construction of Teichmuller spaces. Some notions on the geometry of Teichmuller's spaces.

Riemann surfaces. Uniformization theorem. Complements of sheaf theory and Riemann-Roch theorem. Plane hyperbolic geometry. Hyperbolic geometry of Riemann surfaces. Fundamental domains. Quasi-conformal maps. Extremal maps of Teichmuller. Construction of Teichmuller spaces. Some notions on the geometry of Teichmuller's spaces.

J.H. Hubbard,
Teichmuller Theory and Applications, to Geometry, Topology and Dynamics, vol 1,
Matrix Editions, 2006

Frontal lessons with direct involvement of students in reading parts of the reference text and question and answer sessions with the teacher. Common elaboration of shared notes on the course topics. Development at home of some of the omitted mathematical proofs.

The teacher is available to receive the students for clarifications and to help them in the development of the final seminar, by appointment by e-mail

Professors

RE RICCARDO

Borrowers