33 problems

Let $D$ be a noncommutative division algebra. A free subsemigroup is a multiplicative subsemigroup generated freely by at least two elements. Makar-Limanov's conjecture. The divisi…

Self-similarity conjecture. Assume that $n\geq 2$ and $p\geq d(n^2-1)$. Then no open subgroup of $SL_1^{ne}(D)$ is self-similar.

A finite group $G$ is admissible over a field $F$ if there exists a central division algebra over $F$ containing a $G$-Galois field extension as a maximal subfield. Equivalently, t…

Let $\mathbb K$ be a field, let $n$ be a positive integer, and let $f$ be a multilinear polynomial evaluated on the full matrix algebra $M_n(\mathbb K)$. Kaplansky–L'vov conjecture…

Let $K$ be a field, let $p$ be a prime unequal to the characteristic of $K$, and let $L/K$ be the extension appearing in the relative Brauer group … Write $B_p$ for the $p$-part of…

Let $d$ be a positive integer, let $F$ be an infinite field of characteristic $0$ containing a primitive $d$th root of unity, and let … be a division algebra, where each factor is…

Let $\mathcal{T}$ be the supertranslation algebra associated to a super-Minkowski spacetime, and consider its third Lie algebra cohomology. A 3-cocycle is Lorentz-invariant if it i…

Let $D$ be a division algebra with centre $Z(D)$, and suppose that $Z(D)$ has cohomological dimension $3$ and that $D$ has finite degree over $Z(D)$. Suslin's conjecture. … If true…

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of finite GK dimension. Let $Q(A)$ be its quotient division algebra, and let $D$ be a division s…

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of GK dimension $d$. A chain … of division subalgebras has stratiform length $m$ when each $D_i$…

Conjecture. There is an isomorphism of coalgebras

Computational-effort conjecture. The computational effort for finite fields should be similar for all methods presented in the paper.

Let $K_{\mathrm{UV}}$ denote the ultraviolet value of the scaling quantity $K$ in the renormalization-group trajectory. Ultraviolet fixed-point conjecture. The ultraviolet value is…

Serre's conjecture. The group $\operatorname{SL}_1(\mathcal{O})$ satisfies the subgroup congruence property.

Let $D$ be a finite-dimensional division algebra with center $K$. Its reduced Whitehead group is … where $D^{(1)}=\{x\in D\mid \operatorname{Nrd}_D(x)=1\}$ and $D'=[D^*,D^*]$ for…

Let $K$ be a $C_d$-field for some integer $d>0$, and for each prime $p\in\mathbb{P}$ let $\operatorname{Brd}_p(K)$ denote the Brauer $p$-dimension of $K$. Artin's conjecture. The d…

Noseda–Snopce conjecture. The group $H$ is not self-similar.

Let ${\mathcal D}$ be an index $n$ ${\mathbb Q}$-central division algebra, let $\Lambda\subset {\mathcal D}$ be an order, and let $\psi_{reg}$ denote its regular representation in…

Euler-characteristic conjecture. The integer $\rho$ is the least common multiple of the orders of the finite subgroups of $E$.

Let $A$ be a finite-dimensional division algebra over $\mathbb{Q}$, and let $\Gamma_1,\Gamma_2$ be finitely generated semigroups of the multiplicative group $A^\times$. Fix…

Let $E$ be a field and let $A$ be a central simple algebra over $E$. Its reduced Whitehead group is denoted by $SK_1(A)$. Triviality conjecture. The group $SK_1(A)$ is trivial for…

Let $d_1(r)$ be the piecewise linear function connecting the points $(r,[(m-r)(n-2r)]^+)$ where $2r\in\mathbb{Z}$, and let $d_2(r)$ be the piecewise linear function connecting the…

Let $(X,(U_x)_{x\in X})$ be a proper Moufang set, meaning that its little projective group is not sharply $2$-transitive, and suppose that each root group $U_x$ is abelian. A quadr…

Let $D$ be a finite-dimensional division algebra. Akbari–Mahdavi-Hezavehi conjecture. The multiplicative group $D^*$ has a maximal subgroup, not necessarily a normal one. The quest…

Let $D$ be a division algebra, and let $\operatorname{CK}_{1}(D)$ denote the cokernel of the natural map $K_1(F)\to K_1(D)$, where $F$ is the center of $D$. Ha-Ma-Ha-Mi conjecture.…