On isospectral connections
Abstract: Kac’s celebrated inverse spectral question “Can one hear the shape of a drum?” consists in recovering a metric from the knowledge of the spectrum of its Laplacian. I will discuss a very similar question on negatively-curved manifolds, where the word “metric” is now replaced by “connection” on a vector bundle. This problem turns out to be very rich and connects unexpectedly to two other a priori unrelated fields of mathematics:
1) in dynamical systems: the study of the ergodic behaviour of partially hyperbolic flows obtained as isometric extensions of the geodesic flow (over negatively-curved Riemannian manifolds);
2) in algebraic geometry: the classification of non-trivial algebraic maps between spheres.
Using this relation, I will explain a positive answer to Kac’s inverse spectral problem for connections under a low rank assumption. Joint work with Mihajlo Cekić.