10 problems

Let $G$ be a real semisimple Lie group, $G^{\mathbb C}$ its complexification, and $K^{\mathbb C}$ the complexification of a maximal compact subgroup. For a $G$-orbit $\gamma$ and a…

Let $G_\mathbb R/K$ be a Riemannian symmetric space, let $D$ be the Akhiezer–Gindikin domain, and let $C(S)_0$ denote the cycle space associated with a closed orbit $S$, while…

Bounded-cycle-space fat minor conjecture. For every graph $X$ there exists a function $f:\mathbb{N}^2\to\mathbb{N}^2$ such that, for every graph $G$ whose cycle space is generated…

Let $G$ be an $n$-vertex graph, where $n$ is odd. Write $\kappa(G)$ for its vertex-connectivity, $\alpha(G)$ for its independence number, $\mathcal{C}(G)$ for its cycle space, and…

Generalized $k$-independence conjecture. For every $n$-vertex simple graph $G$,

A graph $G$ is toroidal if it can be embedded on the torus, and its basis number $\operatorname{bn}(G)$ is the smallest integer $k$ for which $G$ has a cycle-space basis in which e…

Barlet's properness conjecture. The restrictions of $P$ to the irreducible components of $\mathcal{C}(\mathcal{X}/\Delta)$ are proper. This is one version of Barlet's properness co…

Let $X$ be an integral projective variety, and let $N_k(X)_{\mathbb{Z}}$ denote the group of numerical classes of $k$-cycles. A class is pseudo-effective if it lies in the pseudo-e…

The complete-partite graph cycle-space conjecture. For every $s\geq 4$,

Let $G$ be a graph. Its hole space is the vector space generated by the holes of $G$ in the cycle space of $G$, and let $k(G)$ denote its competition number. Hole-space dimension c…