10 problems

Bandeira–et-al. degree-three invariant conjecture. If $L\geq 10$, then the generic orbit in $\mathbf{L}^2(S^2)_L$ can be determined up to a finite list of orbits from invariants of…

Cryo-EM sample-complexity conjecture. If $R\geq 3$ is large enough, then the minimal number of observations required for accurate recovery of the $\operatorname{O}(n)$ orbit of $f$…

Let $L$ be a nonnegative integer, let $V=\left(\bigoplus_{\ell=0}^{L}V_\ell\right)^R$ be the corresponding representation, and let the third moment be the invariant obtained by int…

Heterogeneous orbit-recovery conjecture. A phenomenon analogous to the single-signal model should hold: the orbits of multiple signals can be estimated simultaneously from the aver…

Consider MRA with projection for odd signal length $p\geq3$, where reversing the signal produces the chirality ambiguity. MRA-with-projection list-size conjecture. When unique reco…

Let $H=\mathbb{Z}/L\mathbb{Z}\subset\mathrm{SO}(3)$ be generated by rotation through $2\pi/L$ about the $z$-axis, and consider homogeneous projected cryo-EM with $S$ shells and $F$…

Consider heterogeneous projected cryo-EM with $K$ components, $S$ shells, and $F\geq2$ frequencies. Let $U^T_d$ denote the degree-$d$ invariant moment space and let…

Let $\mathcal{F}$ be a finite set of nonzero spherical-harmonic frequencies, let $V$ be the corresponding $\mathrm{SO}(3)$ representation, and let…

Heterogeneous MRA recovery conjecture. Generic unique recovery is possible at degree $3$ if $\mathcal{U}>Kp+K-1$, while generic list recovery is possible at degree $3$ precisely if…

Let $p\geq 3$ be odd, let $V=\mathbb{R}^p$ with the cyclic-shift action of $G=\mathbb{Z}/p\mathbb{Z}$, and let the projection identify opposite entries as in MRA with projection. M…