Classifying real polynomial maps between spheres is a challenging problem in real algebraic geometry. Remarkably, this question has found recent applications in two seemingly unrelated fields:

- in spectral theory, it allowed to solve Kac's celebrated isospectral problem (Can one hear the shape of a drum?) for the connection Laplacian.

- in dynamical systems, it allowed to prove ergodicity for a certain class of partially hyperbolic flows (extensions of the geodesic flow on negatively-curved manifolds).

I will explain these problems and how they all connect together. No prerequisite required -- the talk is intended for a broad audience.

Joint work with Mihajlo Cekić.