3 problems

Weighted Singer Conjecture. The weighted $L^2$-homology vanishes above the middle dimension:

Let $(M,g)$ be a connected, oriented, complete Riemannian manifold whose Ricci curvature is bounded from below. For $t>0$ and $x_0\in M$, let $\rho_t(x,x_0)$ be the heat kernel and…

Let $(W,S)$ be a Coxeter system with nerve $L$, let $Sigma$ be its associated complex, and suppose that $L$ is a generalized homology sphere of dimension $n-1$, written…