4 problems

Sharp weighted Pohozaev–Trudinger constant conjecture. The sharp value of the parameter in the uniform bound for this functional is $\alpha_N$, rather than the currently establishe…

Let $X$ be a finite type space of finite Lusternik–Schnirelmann category. It is rationally hyperbolic when the dimension of $\pi_\ast(X)\otimes\mathbb{Q}$ is infinite, and let…

Let $X$ be a closed manifold, and let $d4db(X)$ denote its free loop space. Its rational homology is $H_\ast(\mathcal{L}(X);\mathbb{Q})$. Gromov's conjecture. The rational homology…

Let $X$ be a space and let $p$ be a prime. Consider the growth of the rational homotopy groups of $X$ and of the $p$-torsion in its homotopy groups. The rational–p-torsion growth c…