5 problems

Helffer–Nier's compact-resolvent conjecture. The operator $P$ is of compact resolvent if and only if its associated Witten Laplacian is of compact resolvent.

Let $\mathbb{M}^{n}(\kappa)$ be an $n$-dimensional complete simply connected Riemannian manifold of constant sectional curvature $\kappa\in\{1,-1\}$, and let $\Omega$ be a bounded…

Let $M$ be a manifold equipped with Morse data and the associated Witten quasimodes, and let the algebraic cup product be represented analytically by the triple products of the cor…

$CD(0,m)$ characterization conjecture. Then the $CD(0,m)$-condition holds, that is,

Let $M$ be a manifold, let $f_i$ be functions on $M$, and let $L_i={\rm graph\ }df_i\subset T^*M$ for $i=1,2,3$. For intersection points $p_{ij}\in L_i\cap L_j$, let $\psi_{ij}$ be…