Veranstaltungsübersicht -- Sommersemester 2020 (Archiv)

For hyperbolic manifolds, there exists a straightforward connection between the spectral and the geometric data. More precisely, the lengths of its closed geodesics and the spectrum of its Laplace operator acting are connected by the Selberg trace formula, that can be considered a sibling of the Poisson summation formula. Selberg trace formula provides the information on the eigenvalues of the Laplace operator, however, completely ignoring its eigenfunctions.

There exists a method, originated from the classical statistical mechanics, that allows to obtain more information on the eigenfunctions. The method, called the transfer operator approach, involves a construction of a so-called transfer operator from a certain discretisation of the geodesic flow on the manifold. For a modular surface, this transfer operator is ultimately connected to a Gauss map. One can show that the 1-eigenfunctions of this operator correspond via a certain integral transform to the eigenfunctions of the Laplace operator. The integral transform mirrors the Eichler-Shimura-Manin isomorphism.

In this talk, we try to construct an analogue of the transfer operator, using the Brownian paths on the manifold instead of the geodesics. We obtain an operator, whose 1-eigenfunctions turn out to be the boundary forms of eigenfunctions of the Laplace operator.