6 problems

Generic semisimplicity conjecture. For the trivial representation $\rho=\rho_{\mathrm{triv}}$, the condition

Faure–Weich–Dolgopyat–Pollicott conjecture.

Deterministic high-frequency gap conjecture. For any $\varepsilon>0$, there is a constant $C_\varepsilon$ such that $\mathbf X_{\mathcal E}$ has no resonance in…

Uniform spectral gap conjecture. For any $\varepsilon>0$, with probability tending to $1$ as $n\to +\infty$, the cover $M_n\to M$ has no new resonance in the half-space

Let $M$ be a compact $(2n+1)$-dimensional manifold and let $\alpha$ be a contact $1$-form on $M$ whose corresponding flow is Anosov with orientable stable and unstable bundles. Let…

Let $\mathbb{M}$ be a negatively curved manifold, let $P_\varepsilon$ be the kinetic Brownian motion operator on $S^*\mathbb{M}$, and let $\nu_\varepsilon>0$ denote the optimal exp…