18 problems

Distinct-zeros conjecture. The positive zeros of $\Sigma(R_1)$ and $\Sigma(R_2)$ are distinct.

Negative-slice conjecture. Outside a single null set, the slice need not be a finite set for every $s>0$; indeed, there may exist a non-trivial measure function $\phi$ such that

Let $Wf(z)$ be the Wigner distribution of some function $f\in L^2(\mathbb{R})$, with bounded nodal set $\mathcal{N}(Wf)$, and suppose that $Wf$ is centered at $z=0$. For a Hermite…

Assume the extremal problem has a unique solution $\varphi_p$, and let $t_1=t_1(p)$ be its first positive zero. First-zero asymptotic conjecture. … and … The source presents these…

Assume the extremal problem has a unique solution $\varphi_p$ for each $p>0$, and let $t_n=t_n(p)$, $n\geq1$, be the positive zeros of $\varphi_p$. Zero-spacing conjecture. … for a…

Assume the extremal problem has a unique solution $\varphi_p$ for each $p>0$, and let $t_n=t_n(p)$, $n\geq1$, be the positive zeros of $\varphi_p$. Zero-monotonicity conjecture. Th…

Let $0<p<\infty$, and let a solution mean a function solving the extremal problem defined earlier in the paper. Uniform-discreteness conjecture. The zero set of any solution of the…

Finite-zero-set conjecture. The zero set of a two-dimensional finite electrical field consists of only finitely many points.

Bounded-zero-set conjecture. The zero set $Z(F)$ is bounded.

Distinct-asymptotic-directions conjecture. The asymptotic directions of the points in the zero sets $X=0$ and $Y=0$, in the Type I and Type II domains, are distinct. Hence, outside…

Polynomial zero-avoidance conjecture. There exists a point $p\in B^d\subset\mathbb R^d$ such that, for every $k=1,\ldots,N$, the point $p$ is at distance at least $\delta_k$ from t…

The planar zero-set conjecture. The zero set of a two-dimensional finite electrical field $F$ consists of a finite set of points.

Let $\theta(q,x)$ be the partial theta function. For $q\in(0,1)$, the zeros are complex-conjugation invariant, so nonreal zeros occur in complex conjugate pairs; for each…

Let $\theta(q,x)$ be the partial theta function, and let a negative real zero mean a real number $x<0$ satisfying $\theta(q,x)=0$. Lower-bound conjecture. Any negative real zero of…

Let $q\in(0,1)$, and let $\theta(q,x)$ be the partial theta function. For each $j\in\mathbb{N}$, let $\Gamma_j$ be the smooth curve formed by the graphs of the zeros $\xi_{2j-1}$ a…

Let $\mu$ be a finite measure with $r_\mu=1$, let $p>0$, and let $\bm{F}(z)=\sum_{n=0}^\infty a_n\zeta_n z^n$ be a Gaussian analytic function as above. Write $A^{p/2}(\mu)$ for the…

Zero-set absolute-continuity conjecture. For every $t\geq 0$, the Slepian zero set on $[0,t]$ is mutually absolutely continuous with respect to the zero set of Brownian motion star…

Let $Z$ be a subset of $B_1\subset\mathbb{R}^n$, and let $F_Z$ be the set of harmonic functions $u$ in $B_1$ such that $Z(u)=Z$ and $u(0)=1$. Compactness conjecture for harmonic fu…