4 problems

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Ghys's Jordan conjecture for diffeomorphism groups of compact manifolds
Ghys's Jordan conjecture. The diffeomorphism group $\operatorname{Diff}(X)$ is Jordan for every compact smooth manifold $X$ .
Conjecturesolvedcompact-manifolds diffeomorphism-groups transformation-groupsarXiv posted 2 days ago
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The dimension bound for compact manifolds with standard lattice actions
Dimension bound conjecture. The dimension of $M$ must be at least $n$ .
Conjectureopencompact-manifolds dynamical-systems lattice-actionsarXiv posted 2 days ago
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Non-Kähler compact manifolds do not admit both SKT and balanced metrics
Let $M$ be a compact complex manifold. A Hermitian metric on $M$ is SKT if its fundamental form $\omega$ satisfies … It is balanced if … where $M$ has complex dimension $n$ . Non-Kä…
Conjectureopencompact-manifolds complex-geometry hermitian-geometryarXiv posted 2 days ago
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The product property for Laplace–Beltrami operators on analytic compact manifolds
Product-property conjecture. This product property holds for Laplace–Beltrami operators on analytic compact manifolds.
Conjectureopencompact-manifolds harmonic-analysis spectral-geometryarXiv posted 2 days ago

Ghys's Jordan conjecture for diffeomorphism groups of compact manifolds