8 problems

Let $\mathcal{A}$ be an $\omega$-categorical structure, let $X$ range over definable sets, and let $\mathbf{K}^X$ denote the associated dual vector space. Let $\overline{X}$ denote…

Strong Extension Principle. Under $\mathrm{OCA}_T$ and $\mathrm{MA}$, the conclusion holds with $U=\emptyset$; equivalently, every such $f$ is the restriction to…

Remainder automorphism conjecture. The remainder $\beta X\setminus X$ has $2^{\mathfrak c}$ algebraically nontrivial autohomeomorphisms.

Let $N$ be a simply connected open $n$-manifold, let $\beta(N)$ be its Stone–Čech compactification, and let $\mathcal S$ be a locally constant sheaf on $\beta(N)$ whose stalk is a…

Let $\beta\mathbb F_2$ be the Stone–Čech compactification of the free group $\mathbb F_2$, and let $M(\beta\mathbb F_2)$ be the Banach algebra of complex, regular Borel measures on…

Let $X$ and $Y$ be locally compact spaces. For a Tychonoff space $T$, let ${\mathscr U}(T)$ be the set of all zero-sets of $\beta T$ that miss $T$, where $\beta T$ is the Stone–Čec…

Maximum-attainment conjecture. If this intersection is nonempty, then every function in $\mathcal{C}$ attains its maximum at every point in the intersection.

Amenability conjecture. If