Séminaires : Séminaire Géométrie et Topologie

Ce séminaire s’adresse aux géomètres, topologues et dynamiciens au sens large. Il est rattaché aux équipes Analyse Algébrique et Analyse Complexe et Géométrie. Les exposés seront accessibles à une audience large, doctorants inclus. Il se tiendra à Jussieu, le jeudi à 11h, en salle 15-25 502. Le séminaire a l'agenda google suivante: https://calendar.google.com/calendar/b/0?cid=dDgzNTJoczNmdDhlMm5nb2IzMXJwaWpsdHNAZ3JvdXAuY2FsZW5kYXIuZ29vZ2xlLmNvbQ

Some years ago, Bruggeman, Lewis and Zagier provided a cohomological interpretation of Maass wave forms for hyperbolic surfaces of finite area, in which each construction and isomorphism is explicit. About the same time, I developed discretizations for geodesic flows on a huge class of hyperbolic surfaces such that the 1-eigenfunctions of the associated transfer operators serve as building blocks for the cocycle classes in these cohomological interpretations. By combining both results we find an explicit and constructive relation between the geodesic flow and Maass cusp forms for non-compact hyperbolic surfaces of finite area.

Whereas the construction of the discretization of the transfer operators was valid also for hyperbolic surfaces of infinite area, the question on suitable cohomological interpretations of Laplace eigenfunctions in infinite area situations remained open. Only recently Bruggeman and I provided a generalization of the cohomology to (a class of) hyperbolic surfaces of infinite area that serves in the same way as in the finite area case as a mediator between Laplace eigenfunctions and eigenfunctions of transfer operators.

I will survey these new results with an emphasis on insights, heuristics and their relation to dynamics.