6 problems

Let $M^n$ be a closed Riemannian manifold satisfying … Here $C_{P,k}$ denotes the Poincaré constant for $k$-forms. Sectional-curvature spectral gap conjecture. For every $k\ge 1$,…

Let $M^n$ be a closed Riemannian manifold with … Here $C_{P,1}$ denotes the Poincaré constant for $1$-forms. Honda–Mondino's spectral gap conjecture. Under these assumptions, … Thi…

Let $({\cal A},h)$ be a nonlinear filtering model, let $\pi_t^\nu$ denote its conditional filter, and let the standard Poincaré inequality for the underlying Markov process be … He…

Let $G$ be a Carnot group and let $N$ be a smooth homogeneous norm on $G$. For a function $U=U(N)$, write $\mu_U$ for the corresponding probability measure with density $e^{-U}$ wi…

Exponential-concentration Poincaré conjecture. There exists a constant $C(M)$ such that

Weighted orthogonality Poincaré conjecture. One has