Course Unit Page

  • Teacher Stefano Francaviglia

  • Learning modules Stefano Francaviglia (Modulo 1)
    (Modulo 2)

  • Credits 6

  • SSD MAT/03

  • Teaching Mode Traditional lectures (Modulo 1)
    Traditional lectures (Modulo 2)

  • Language English

  • Campus of Bologna

  • Degree Programme Second cycle degree programme (LM) in Mathematics (cod. 5827)

Academic Year 2022/2023

Learning outcomes

The student will learn the theory of Teichmuller spaces. The main viewpoint will be that of abelian differentials on Riemann surfaces and traslation surfaces, with links to hyperbolic surfaces. Some dynamical aspects such as geodesic and horocyclic flows will be studied in detail.

Course contents

Geometry of the hyperbolic plane in the half plane model and action of SL(2,R).  Geodesic flow and horocyclic flow. Fuchsian groups and their quotients. Teichmuller space. Ergodic theory and invariant measures. Introduction to Ratner theory.


Einsiedler, Ward: "Ergodic Theory with a view towards number theory", Springer, 2011
Beardon: "The geometry of discrete groups", Springer, 1983.

Teaching methods

Classical frontal lessons

Assessment methods

Oral exam

Teaching tools

Use of online platforms as Virtuale, with dedicated forums, online material etc.

Office hours

See the website of Stefano Francaviglia

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