Title of the course: Introduction to hyperbolic geometry
Instructor: Dr. Daniel Massart
Institution: Institut Montpelliérain Alexander Grothendieck, Montpellier, France
Dates: 18-24 September 2023
Prerequisites: Complex numbers, linear algebra (2×2 matrices)
Level: 1st year undergraduate
Abstract: We begin with the Poincaré half-plane and disc models of hyperbolic geometry. We determine the isometry group, and the geodesics (the straight lines, if you will). We explain Poincaré’s geometric construction of discrete subgroups of the isometry group, allowing us to construct surfaces of infinitely many topological types, whose geometry is hyperbolic. This proves that hyperbolic geometry is richer than Euclidean geometry, since the only complete Euclidean surfaces are the plane, the cylinder, the Möbius strip, the torus, and the Klein bottle. We also prove that the space of all tori (or moduli space of elliptic curves-don’t worry if you have never heard any of these words) has a natural geometry, and this geometry is hyperbolic, so in a sense hyperbolic geometry rules over Euclidean geometry.
Language: EN