74 problems

Unique-centre conjecture. If a simplex has a unique simplex centre, then it is equifacetal.

Let $P_n(x)$ be the generating function for alternating sign matrices invariant under flips in the vertical and horizontal axes, with the weight defined in the source. Such matrice…

Let $Q_n(x,y)$ be the generating function for quarter-turn-symmetric alternating sign matrices. Quarter-turn polynomial factorization conjecture. There exists a sequence of polynom…

Let $Q_n(x,y)$ be the generating function for quarter-turn-symmetric $n$ by $n$ alternating sign matrices, and let $H_n$ and $A_n$ denote the half-turn-symmetric and unrestricted a…

Let $H_n(x,y)$ be the generating function for half-turn-symmetric alternating sign matrices, and let $R_n(x,\mu)$, $T_n(x,\mu)$, and the polynomials $S_{4n+1}(x),S_{4n-1}(x)$ be as…

Let $H_n(x,y)$ be the weight-generating function for half-turn-invariant $n$ by $n$ alternating sign matrices, and let $Z_n(x,y,0)$ and $Z_{n-1}(x,y,1)$ be the corresponding plane-…

Vertically symmetric ground-state conjecture. The vector

Half-turn-symmetric ground-state conjecture. The vector

Symmetric capacity conjecture. The symmetric capacity satisfies

Consider real matrix ensembles with additional linear constraints, including real symmetric or other structured real matrices. Let $\rho(\lambda)\,dx\,dy$ denote an absolutely cont…

Let an effective physical model be a model featuring an isometry, and let all fields be invariant under this isometry. Suppose the model is mapped by a reference system $S$ satisfy…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a strictly convex body, and let $\mathcal S$ be a family of hyperplanes that is the image of an embedding of the tangent bundle of the sphe…

Multiple minimizers via symmetry. The constrained problem admits at least three distinct minimizers related by rotations through $2\pi/3$.

Let $n \geq 5$ be such that $n \not\equiv 2 \pmod 4$, and set $k = \floor*{n/2}$. The polytope $P_\mathrm{QTSASM}$ is the polytope of quarter-turn symmetric alternating sign matric…

Move-connectivity conjecture. If

Let $K\subset \mathbb{R}^{n}$ be a convex body with $n\geq 3$, and let $p\in\operatorname*{\int}K$. For every hyperplane $\Pi$ passing through $p$, suppose the section $\Pi\cap K$ h…

Let $K\subset \mathbb{R}^3$ be a convex body. A plane section means the intersection of $K$ with an affine plane, and an axis of symmetry is a line in that plane whose reflection p…

Berestycki–Caffarelli–Nirenberg conjecture. (i) If there is a positive bounded solution $u$ of this problem, then $u$ depends only on $x_n$; consequently, $u_{x_n}>0$ and…

Reflection-symmetric strengthening. There exists a set $\Pi=\{\pi_1,\pi_2,\ldots,\pi_{C_k}\}$ of $C_k$ permutations of $\mathbb{Z}_{2k+1}$, each fixing $0$, and a bijection…

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body, and let $G<\operatorname{O}(\mathbb R^n)$ be its symmetry group. Let $\pi_k$ and $\rho_k$ denote the subgroup types defined…

Alvino–Ferone–Trombetti's conjecture. Every extremal function $u$ for inequality (1.1) is of the form

Behrend–Fischer–Koutschan symmetry conjecture. Given $n$, for any $\rho$ and $\tau$, we have

Grad's conjecture. If there exists an MHD equilibrium $(X,p)$ on $M$ whose pressure has toroidally nested level sets, then the equilibrium is axisymmetric.

Let $X^{\mathrm O}_{2n}(r,t)$ be the even-order OSASM generating function, and let $\bar t$ denote the parameter appearing in the source's generating-function notation. For an OSAS…

Rotational-invariance conjecture. If the activation function $csigma$ is a smooth function that is not linear and the data set contains $cgeq 2$ data points, then