28 problems

Pataki's conjecture. Facial exposedness and niceness are equivalent: every facially exposed closed convex cone is nice.

Finite-generation conjecture. For every positive integer $n$, this closed cone is finitely generated.

Cone equality conjecture. The two cones are equal:

Boundary-stratum conjecture. There exists a subcone $S+\sigma\subset S$ such that

The bounded dual-cone generators conjecture. $G$ is a semigroup basis for $S_\sigma$.

Let $A\in\mathbb{R}_{\geq 0}^{m\times r}$ and consider the cone toolkit, in which rows of $A$ determine facets of the associated cone and its extreme rays provide candidate vectors…

Let $\mathcal{P}_{\mathrm{PPT}}$ be the PPT cone, let $\operatorname{MCL}_k$ denote the rank-constrained $C^*$-convex hull operation, and set … Let $k_c$ be the critical rank for f…

Let $\overline M_{0,n}$ be the moduli space of stable genus-zero curves with $n$ marked points. Let $\operatorname{PBDcone}(n)$ be the cone associated with effective partition bloc…

Let $Q\subseteq\mathbb R^n$ be a polyhedral cone, and let ${\rm MA}(P,Q)$ denote the problem of computing the maximal angle between cones $P$ and $Q$. Orthant-restriction conjectur…

Survival-probability convergence conjecture. The conclusion of Theorem concerning the asymptotic survival probability remains valid for cones that are not pyramids, provided the pa…

Sebő's conjecture. The ICP holds for every $n$-dimensional cone $C$.

Let $\operatorname{\textsf{Cones}}(d)$ be the partially ordered set of pointed, rational, finitely generated cones in $\mathbb{R}^d$, ordered by successive addition of nonnegative…

Inclusion conjecture.

Critical-exponent conjecture. If $mathcal K^o ot itext{?}$ is an open convex cone with $mathcal K^o ot itext{?}$ and $mathcal K^o ot itext{?}$, then there exists

16-source generation conjecture. Each source may be written as a positive linear combination with integer coefficients of index vectors taken from these 16 sources:

Let $K={\bf R}_+^d$ with $d\geqslant 3$, and consider the moment conditions in the first claim of Theorem, which require finiteness of a moment of order $p+q_{\sigma}+d-2+(2-p)^+$…

Let $p\in \mathbb{R}[x_1,\dots,x_n]$ be a complete hyperbolic polynomial with hyperbolicity cone $C$. For a real $\alpha>n$, let $q_{\alpha}$ be the function defined by … where…

Non-SAGE separation conjecture. If every $\bm{a}_i$ lies in either $\operatorname{ext}\mathcal{P}(\bm{A})$ or $\operatorname{int}\mathcal{P}(\bm{A})$, but $\bm{A}$ does not satisfy…

Let $Q:{\mathbb R}^n\to{\mathbb R}^m$ be an elliptic quadratic operator with $\textup{rg}Q=2$, and let ${\overline K}'_Q$ denote its associated cone. Let $C_3$ be a three-dimension…

Pseudo-effective and movable cone equality conjecture. The equality holds

Let $\Omega$ be an open bounded convex domain, and let a cone over $\Omega$ mean an open proper convex cone whose projectivization contains $\Omega$. A non-projective isometry is a…

Let $\mathcal{C}_n$ denote the cone of copositive matrices of order $n$, and let $\theta_{\max}(\mathcal{C}_n)$ be the maximal possible angle between two matrices in this cone. Hir…

Let $C(\operatorname{SqFree},n)$ be the positive rational cone generated by homological data triplets of free squarefree complexes over $\Bbbk[x_1,\ldots,x_n]$. A pure free squaref…

Let $S$ be a finite spanning set, and let $\mathfrak M_\mathbb Z(S)$ be the lattice generated by standard measures of simplices with vertices in $S$. Define the rational cone…

Limiting-cone equality conjecture.