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.