Common-core 1 and 2- Geometry of the hyperbolic disk: isometries, geodesics, alternative models.
- Action of the modular group: fundamental domain, Poincaré’s paving theorem.
- Fuchsian groups.
- Statement of the uniformization theorem.
Common-core 3 and 4- Definition of modular forms.
- Construction of Eisenstein series and Poincaré series.
- The Delta form and the algebra of modular forms in the case of SL(2,Z).
- The j-invariant.
- Periodic geodesics on the modular surface, dense geodesics, relation to the Markov spectrum.
- Ergodicity of the geodesic flow.
- The Hower-Moore theorem in the case of SL(2,R).
Number theory 1 and 2- The algebra of modular forms on SL(2,Z) and the algebra of mod p modular forms.
- Application to Kummer congruences.
Number theory 3 and 4- p-adic numbers.
- Serre’s p-adic modular forms.
- Application to the construction of the Kubota-Leopoldt p-adic zeta function.
Number theory 5- Class fields of cyclotomic fields, introduction to the Iwasawa main conjecture and the Mazur-Wiles theorem.
Geometry and dynamics 1 and 2- Topology of SL(2,R)/SL(2,Z), relation to the Milnor fibration.
- Crash course on knot theory and discussion of singularities of complex algebraic curves.
Geometry and dynamics 3 and 4- Relation to the Markov spectrum.
- The Dedekind eta function and its relation to geodesic entanglement.
Geometry and dynamics 5- Introduction to the Lorenz attractor and its relation to modular knots, periodic geodesics on the modular surface.
