Math 100: Topics in Probability

Peter Doyle

Dartmouth College
Spring 2007

The class meets Mondays and Wednesdays at 4:00 in 201 Kemeny Hall.

The subject for the quarter is `A Brownian approach to spectral geometry'. We will use the theory of Brownian motion (a polite term for `drunkard's walk') as a source of tools and insight into problems in spectral geometry. A list of some possible topics is given below.

The geometry of diffusion

• Drunkard's walk
• Brownian motion on the line
• Brownian motion in higher dimensions
• Transience and recurrence
• Laplace's equation
• The heat equation
• Brownian motion on manifolds
• Conformal mapping
• Brownian motion on Riemann surfaces
• The type problem
• Shorting and cutting
• Ergodicity of geodesic flow
• Eigenvalues
• The wave equation
• The free Gaussian field

Spectral geometry

• Spectral invariants
• Isospectrality
• Unions of flat tori
• Drums
• Drumsticks
• Dirichlet versus Neumann
• Homogeneous geometries
• The method of images
• The Selberg trace formula
• The counting trace
• Accidental isospectrality
• Scenarios
• Tetra and Didi, the cosmic spectral twins
• Hyperbolic 2-manifolds
• Hearing conepoints
• Disconnected flat 2-orbifolds
• The Prime Geodesic Theorem
• Hyperbolic 2-orbifolds
• Lens manifolds and lens orbifolds

Diffraction

• The wave equation
• Reflection and diffraction
• Diffraction by an infinite half-plane
• Harmonic sources
• Repeated diffraction
• Waves in a polyhedron