50 problems

Let $X\sim f_X$ and $Z\sim f_Z$ be independent random variables, where $f_X$ and $f_Z$ are densities with cumulative distribution functions $F_X$ and $F_Z$, respectively. Integral…

Let $X$ and $Z$ be independent draws from arbitrary unknown probability distributions with densities $f_X$ and $f_Z$, satisfying … Let $Y$ be drawn from the mixture distribution ……

Classification conjecture. Any MUB-triplet $(X_1,X_2,X_3)$ in dimension $d=6$ is permutationally unitary equivalent to an MUB-triplet obtained by the construction $(\mathrm{hff})$.…

Classification conjecture. Every $2$-dimensional Leibniz triple system over $F$ is isomorphic to one of the systems 1–5.

Let $A$ be a simple, unital, separable, and nuclear $C^*$-algebra of stable rank one. Write $V(A)$ for its Murray–von Neumann semigroup of projections and $[1_A]$ for the class of…

Let $M$ be a connected properly immersed minimal surface in ${\mathbb R}^3$, and let $A(M)$ denote its area growth constant. Let $S_\theta$, for $\theta \in (0,\frac{\pi}{2}]$, den…

Let $E$ and $F$ be finite graphs, let $\mathsf{k}$ be a field, and write $L_{\mathsf{k}}(E)$ and $L_{\mathsf{k}}(F)$ for their Leavitt path algebras. Their graded Grothendieck grou…

Let $F$ be a field of zero characteristic, and let $J$ be a simple Jordan superalgebra over $F$ with nonzero odd part. The possible superalgebras include $H(A,*)$, where $(A,*)$ is…

Toric 2-Fano classification conjecture. The only toric 2-Fano manifolds are projective spaces.

The Toms–Winter conjecture. The following conditions are equivalent:

A $C^*$-algebra is called classifiable if it belongs to the class of algebras classified by its Elliott invariant, and a groupoid model is HK-good when it has the homology–$K$-theo…

Let $E$ and $F$ be finite graphs, and let $K$ be a field. For a … -graded case with its natural $\mathbb{Z}[x,x^{-1}]$-module structure. Let $T_E$ denote the talented monoid of $E$…

Let $\overline{X}$ be a singular del Pezzo surface. The height bound conjecture. If $\operatorname{char}\mathscr{k}\neq 2,3$, then … The height is a bounded invariant intended to o…

Let $K^{\lor}$ be the spherical completion of the underlying valued field $K$, let $V_K$ denote the relevant multiplicative value subgroup, and let $\sim$ denote the equivalence re…

Chain-length characterization conjecture. 1. If

Let $x$, $y$, and $z$ be three points on the real line, where $x$ and $z$ are generated independently from two distinct distributions and $y$ is generated from their 50--50 mixture…

Equivariant classification conjecture. There is an order-preserving $\mathbb Z[x,x^{-1}]$-module isomorphism

Equivariant K-theory classification conjecture. There is an order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism

Hazrat's fullness conjecture. There exists a unital $\mathbb{Z}$-graded $\mathsf{k}$-homomorphism $\psi:L(E)\rightarrow L(F)$ such that

Graded classification conjecture. There is an order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism

Self-duality conjecture. The class feature aggregate is proportional to its corresponding classifier weight: for each $k=1,\ldots,K$, $\boldsymbol{h}_k^*=\alpha\boldsymbol{w}_k^*$…

Within-class collapsing. All features within each class are identical: for each $k=1,\ldots,K$, $\boldsymbol{h}_{k,i}^*\equiv\boldsymbol{h}_k^*$ for every $i$.

Killing-field classification conjecture. The conclusion of the Hermitian classification conjecture holds for $(\mathcal{M},g_{ab})$.

Hermitian classification conjecture. $(\mathcal{M},g_{ab})$ is one of Kerr, Chen--Teo, Taub-bolt, or Taub--NUT with the orientation opposite to the hyperkähler orientation.

Let $A$ be a unital, separable, nuclear ${\rm C}^*$-algebra of real rank zero and stable rank one, and let $\underline{\mathrm{Cu}}_u(A)$ denote its total Cuntz semigroup invariant…