We characterize the tempered part of automorphic Langlands category D-mod(Bun_G) using geometry big cell in affine Grassmannian. deduce that, for $G$ non-abelian, D-modules have no de Rham cohomology with compact supports. The latter fact boils down to a concrete statement, which we prove Ran space and some explicit t-structure estimates: non-abelian $\Sigma$ smooth curve, Borel-Moore homology ...

We relate the machinery of persistence modules to Legendrian contact homology theory and Poisson bracket invariants, use it show existence connecting trajectories symplectic Hamiltonian flows.