Modul: MAT675 PDE and Mathematical Physics

## Holonomy Inverse Problem

Talk by Dr. Mihajlo Cekić

**Date:** 28.10.21 **Time:** 18.00 - 19.30 **Room:** Y27H35/36

Given a compact Riemannian manifold (M, g) and a vector bundle over M equipped with a connection, we consider the following question: does the holonomy along closed geodesics determine the gauge (equivalence) class of the connection? If (M, g) has negative curvature or more generally its geodesic flow is Anosov, in this talk I will explain how in fact, only the traces of the holonomy along closed geodesics locally determine a generic connection; global uniqueness results are obtained in some cases. A direct consequence is an inverse spectral result for the connection (magnetic) Laplacian. The proof relies on two new ingredients: a Livšic type theorem in hyperbolic dynamics for unitary cocycles, and the interplay between the local geometry of the moduli space of connections with Pollicott-Ruelle resonances of a certain natural transport operator. Joint work with Thibault Lefeuvre.