Entry requirements

60 credit points in mathematics.

Aims

The course gives basic knowledge about hyperbolic

geometry in two dimensions, primarily regarding

discrete groups of isometries and it

also introduces automorphic functions and forms.

The course should also

give examples of applications of automorphic

functions within for

example analytic number theory and quantum chaos.

Content

Hyperbolic geometry: isometries and geodesics,

the disc- and the upper

half-planemodel, metric and measure,

the Gauss-Bonnet formula.

Discrete groups: Fuchsian and Kleinian groups, fundamental domains,

Siegels theorem, the limit set.

Introduction to classical (algebraic) Riemannsurfaces and

uniformisation.

Elliptic functions: Weierstrass P-function, the

modular function

lambda, the little Picard theorem.

Introduction to automorphic functions and forms. Poincaré series

and

other examples of automorphic functions.

Applications in analytic number theory.

Introduction to non-holomorphic (Maass) automorphic functions with

applications in quantum chaos.

If there is an interest and time left some digressions can be made

in

for example spectral, analytic number theory, arithmetical fuchsian

groups and quaternion algebras or the connection between

automorphic

functions and representation theory (this requires some

prerequisites in

algebra).

Instruction

Lectures and problem solving sessions.

Assessment

The examination will be in the form of written

and/or oral exam.

Mandatory assignments may occur.