6 problems

Connectedness conjecture. The closures of ${\cal V}$ and ${\cal W}$ should be connected without the assumption that there are irreducible tuples.

Separation conjecture. If g\text{in \hspace{-.8em}/}\langle\alpha,\beta\rangle, then $\alpha$ and $\beta$ lie in the same connected component of

Let $R$ be a real closed field. For a smooth algebraic variety $X$ over $R$, write $X^{\operatorname{h}}$ for its associated space of real points. A generalized connected open subs…

Let $\mathcal{C}$ be a tuple of semisimple conjugacy classes in $\operatorname{GL}_n\langle\sigma\rangle$, not necessarily generic, and suppose that the character variety…

Shokurov–Kollár connectedness conjecture. For every point $s\in S$, the fiber of the non-klt locus has at most two connected components:

Let $A\in M_2(\mathbb{Z})$ be expanding with $|\det(A)|=3$, let $f$ be its characteristic polynomial, and let ${\mathcal{D}}=\{0,1,b\}v$ with $b>1$ and $v\in\mathbb{R}^2$ such that…