(J Funke and J R Parker)
DescriptionThe modular group Γ=PSL(2,Z) is the group of Möbius transformations with integer coefficients and determinant 1. That is, the group of all transformations T(τ)=(aτ+b)/(cτ+d) where a, b, c, d are integers with ad-bc=1.There are two index 6 subgroups of the modular group. One of these is the fundamental group of a torus with a hole and the other of a sphere with three holes. The natural geometry of the torus with a hole or the sphere with three holes is hyperbolic geometry. This leads to a deep connection between hyperbolic geometry and number theory, in particular the Markoff equation x2+y2+z2=xyz. The algorithm for generating continued fractions of real numbers also allows us to write elements of the modular group in terms of the generators T_1(τ)=τ+1 and T_2(τ)=-1/τ. This in turn leads to a connection between simple curves (that is curves that do not intersect themselves) on the torus with a hole and the sphere with three holes. In this project you will explore the connections between continued fractions, the modular group and hyperbolic geometry. It may be helpful if you have taken Geometry III or Elliptic Functions III. The project will be supervised by Jens Funke in Michaelmas term and John Parker in Epiphany term.
ResourcesA good introduction to the topic is in the paper The geometry of the Markoff numbers by Caroline Series, published in The Mathematical Intelligencer 7 (1985), no. 3, 20–29. |