6 problems

Let $M$ be a closed Riemannian manifold, let $\widetilde{M}$ be its universal cover, and let $\operatorname{Isom}(\widetilde{M})$ be its isometry group containing $\pi_1(M)$. Quant…

Let $(M_1,g_1,J_1,\omega_1)$ and $(M_2,g_2,J_2,\omega_2)$ be compact Kähler manifolds. Let $\operatorname{Isom_{\mathrm{Kah}}}(M_i)$ denote the Lie group of holomorphic isometries of…

Let $\overline{N}\cong \mathcal{H}\times L$ be the q-map space, with $g_{L}(p)$ the left-invariant metric on the fiber $L_p\cong L$ parametrized by $p\in\mathcal{H}$. Let $G$ be th…

Farb and Weinberger's conjecture. 1. $M$ is aspherical, smoothly irreducible, $\pi_1(M)$ has no non-trivial normal abelian subgroup, and

Let $X$ be a complex HI space, meaning a hereditarily indecomposable Banach space, and let the group of isometries of $X$ act on $X$. Near-triviality conjecture. The group of isome…

Let $G$ be a group and let $X$ be a real Banach space with $\dim X \geqslant |G|$. An equivalent renorming of $X$ is a norm equivalent to its original norm, and…