8 problems

Let $\mathrm{SO}(n)$ be the group of $n\times n$ rotation matrices, and let Kac's walk on $\mathrm{SO}(n)$ update by multiplying the current state by a uniformly chosen coordinate-…

Multidirectional scaling conjecture. For $d>3$, crossing the identity-root manifold generically requires scaling in $\lfloor d/2\rfloor$ independent directions, each governed by it…

Matsumoto's trajectoid return conjecture. Such bodies exist for almost any path, with the body returning to its original orientation after two or more periods.

Let $\mathrm{SO}(3)$ denote the rotation group. A quadrature formula on $\mathrm{SO}(3)$ with equal weights is said to reproduce integrals of polynomials of degree at most $2$ exac…

Dense-orbit conjecture. If ${\sf X}_1$ does not conform rationally to ${\sf X}_2$, then $P{\cal F}$ is dense in $\Re^3$ for all $P\in\Re^3\setminus({\sf X}_1\cup{\sf X}_2)$.

Rational-conformity conjecture. $P{\cal F}$ is nowhere dense for each $P\in\Re^3$ if and only if ${\sf X}_1$ conforms rationally to ${\sf X}_2$.

Hexagonal-tiling conjecture. If plane ${\sf H}\supseteq\{Pf,Pfr_{AB},Pfr_{AC}\}$, then ${\sf H}\cap P{\cal G}$ is the vertex set of a hexagonal tiling of ${\sf H}$.

Let $\varGamma=\langle\sigma_1,\sigma_2,\ldots,\sigma_t\rangle$ be a finitely generated subgroup of $SO_3(\mathbb{R})$. Define the self-adjoint averaging operator on $L^2(S^2,dA)$…