75 problems

For every finite-dimensional representation-finite algebra $A$ and every integer $i\ge 0$, the number of quotient-closed full additive subcategories…

Let $\Phi$ be a Leonard system with primitive idempotents $E_0,E_1,\ldots,E_d$ and dual primitive idempotents $E^*_0,E^*_1,\ldots,E^*_d$ in its associated algebra $\mathcal A$. The…

Commutative loop conjecture. The structure $(3^{A},\oplus)$ satisfies:

Let $(\mathcal A,*)$ be an associative, unital, finite-dimensional $*$-algebra over $\mathbb R$. The spectral monoid $\overline{\operatorname{Her}}(\mathcal A,*)$ is the monoid of…

Generalized Kaneko–Zagier conjecture. For each $\alpha\in(\mathbb{Z}/N\mathbb{Z})^\times$, the $\mathbb{Q}(\zeta_N)$-linear map

The commutative loop conjecture. For all $X\in 2^{IN}$, $(3^X,\oplus)$ satisfies: (1) $(\forall A,B\in 3^X)(A\oplus B\in 3^X)$; (2)…

Extended double-shuffle conjecture. The induced map $\varphi_R$ is injective; equivalently, the algebra of Euler sums is isomorphic to $R_{\mathrm{EDS}}$.

Let $q$ be a prime power, let $\mathbb{F}_q$ be the field with $q$ elements, and let $\overline{\mathbb{F}_q}$ denote an algebraic closure of $\mathbb{F}_q$. For…

For a number field $K$ and an abelian variety $A$ over $K$, let $A(K)$ denote the group of $K$-rational points of $A$ and let … be its torsion subgroup, which is finite since…

In mathematics, the Suita conjecture is a conjecture related to the theory of the Riemann surface, the boundary behavior of conformal maps, the theory of Bergman kernel, and the th…

Let $K/k$ be a finite Galois extension of number fields with Galois group $G$, and let $S$ be a finite set of places of $k$ containing all archimedean places and all places ramifie…

Let $K$ be a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings, let $\mathcal{O}_K$ be its ring of integers, and let $E = \mathcal{O}_K^{\times}$ be its…

Let $p$ be an odd prime, let $\zeta_p$ be a primitive $p$-th root of unity, and let … be the maximal real subfield of the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$. Let $h_K$ d…

Let $K$ be an algebraic number field (a finite extension of $\mathbb{Q}$) with a fixed algebraic closure $\overline{K}$, and let … be the maximal abelian extension of $K$. For ever…

Let $K$ be an algebraic number field, i.e. a finite extension of $\mathbb{Q}$, and let $\mathcal{O}_K$ denote its ring of integers. For $n \ge 1$, consider the quadratic forms … wi…

Let $K$ be an algebraic number field with adele ring $\mathbf{A}_K$, and let $L/K$ be a finite Galois extension with Galois group $G=\mathrm{Gal}(L/K)$. Let … be an irreducible com…

Call a sequence $(a_n)_{n\ge 0}$ of natural numbers eventually periodic if there exist $N\ge 0$ and $p\ge 1$ such that $a_{n+p}=a_n$ for every $nge N$. Call a real number $x$ *alge…

Let $\ell$ be a prime number and let $k$ be a finite extension of $\mathbb{Q}$. Inside the field $k(\mu_{\ell^{\infty}})$ generated by all $\ell$-power roots of unity there is a un…

Let $E/k$ be a quadratic extension of local fields with $k$ not of characteristic $2$, and let $\varepsilon \in \{+1,-1\}$. Let $V$ be a finite-dimensional $E$-vector space equippe…

Fix a prime $p$ and an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$, and let $G{\mathbb{Q}} = \mathrm{Gal}(\overline{\mathbb{Q}},|,\mathbb{Q})$. Let…

For a squarefree integer $d \neq 1$ put $K=\mathbb{Q}(\sqrt{d})$, let $\mathcal{O}_K$ be the ring of integers of $K$, and let … be the order of the ideal class group of…

Let $k$ be a field of characteristic zero and let $R$ be a finitely generated $k$-algebra. For a commutative $k$-algebra $A$, write … regarded as an $A$-module via…

A wild problem is a class of problems where the various parts that specify the problem can change, or keep changing, and there is no single "right" outcome to aim for. Every move o…

Let $K$ be a number field and let $g ge 2$ be an integer. For a smooth projective geometrically connected curve $C$ defined over $K$, write $C(K)$ for the set of $K$-rational point…

Let $R$ be a commutative Noetherian regular local ring, let $\dim$ denote Krull dimension, and for an $R$-module $M$ let $\ell_R(M)$ denote its length. Let $P$ and $Q$ be prime ide…