124 problems

Let $A$ be a simple separable unital nuclear $C^*$-algebra. The Toms–Winter conjecture. Finite nuclear dimension of $A$, tensorial absorption of the Jiang–Su algebra $\mathcal{Z}$,…

Elliott's classification conjecture. Unital separable simple nuclear $C^*$-algebras are classified up to isomorphism by $K$-theoretic invariants.

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…

Frenkel–Lepowsky–Meurman uniqueness conjecture. The moonshine VOA $V^\natural$ is the unique holomorphic VOA of central charge $24$ with $\dim V_1=0$.

Let $K_0^{\mathrm{gr}}$ denote the graded Grothendieck group of a graded algebra. For Leavitt path algebras and graph $C^*$-algebras, the relevant graded Grothendieck group is inte…

Let $\Kee$ be an algebraically closed field of characteristic $p>7$, and consider simple finite-dimensional restricted Lie algebras over $\Kee$. Kostrikin–Shafarevich conjecture. T…

The three regularity properties of $\mathcal{Z}$-stability, finite nuclear dimension, and the relevant comparison property of positive elements are considered for separable simple…

Wang–Zhang–Zhuang's classification conjecture. The families (I)–(VI) constitute all affine Hopf $k$-algebra domains of Gelfand–Kirillov dimension $2$, at least in the pointed case.

The conjecture. Every finite-dimensional composition $\mathcal{Z}_2$-algebra is isometrically isomorphic to one of the following ten $\mathcal{Z}_2$-algebras:

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…

Robert's classification conjecture. The scaled cone of lower-semicontinuous traces classifies separable nuclear $C^*$-algebras up to $a$-tensorisation.

Solomon's conjecture. Every simple $2$-fusion system is either a Benson–Solomon fusion system or the $2$-fusion system of some finite simple group.

Let $A$ be a simple, unital, separable, nuclear, infinite-dimensional C$^*$-algebra. Winter–Tikuisis regularity conjecture. The following are equivalent: 1. $A$ has finite nuclear…

Centralizer characterisation conjecture. If $C\cong \operatorname{GL}_2(K)$, then $G\cong \operatorname{PSL}_3(K)$; if $C\cong \operatorname{PSL}_2(K)\times K^*$, then…

Let $A$ be a simple, separable, nuclear approximately subhomogeneous $C^*$-algebra with strict slow dimension growth. The algebra $A$ is $\mathcal{Z}$-stable when…

Let $\mathbb K$ be an algebraically closed field of characteristic $p>0$. For a simple finite-dimensional complex Lie algebra $\mathfrak g$, let $\mathfrak g_{\mathbb Z}$ be a…

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…

Dimension-$p^3$ classification conjecture. Every Hopf algebra $H$ of dimension $p^3$ is semisimple, pointed, or has pointed dual; equivalently, $H$ is one of the Hopf algebras in t…

Let $M$ be a connected compact smooth manifold with $\dim(M)>0$, and let $h\colon M\to M$ be a uniquely ergodic minimal diffeomorphism. Let…

Elliott's trace approximation conjecture. Every trace on $A$ has that strongest approximation property:

Kummer conjecture. Every simple compact Kähler threefold is Kummer.

Let an infinite series of anticanonically embedded quasi smooth Fano hypersurfaces in weighted projective 4-spaces be given. The series in the known classification have the form ……

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

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…