12 problems

Let $n$ be a positive integer and let $a,b\in\mathbb{C}$ satisfy $|a|\neq 1$. Set $h(z,w)=z^n+aw^n+b$ and $h^{\ast}$ for the polynomial obtained by conjugating the coefficients and…

Let the coefficients of the harmonic polynomial model be independent and identically distributed random variables with mean zero, positive variance, and finite moments. The i.i.d.…

Let $d_j:\Omega_n\to\Omega_n$ be defined for $j\geq1$ by … and let $\delta_n\in\mathbb{C}[x_1,\dots,x_n]$ be the Vandermonde determinant. Let $SR_n$ be the superspace coinvariant a…

Let $H_{n,m}(z)=p(z)+q(\overline{z})$ be a random harmonic polynomial in the Weyl model, with $p$ and $q$ analytic complex polynomials of degrees $n$ and $m$, respectively. Let…

Let $p\in\mathbb{Z}^+$, define $p^*=p\mod 9$, and let $n_0(p)$ denote the first value of $n$ for which the two additional solutions in the paper's theorem occur. Threshold formula…

Lee–Lerario–Lundberg bound. The number of solutions of this equation is bounded by

Let $S\subset\mathbb R^d$. Call $S$ an injectivity set for the spherical mean transform if vanishing of all spherical means centered on $S$ forces the underlying function or distri…

Let $\mathscr{H}_n^{(r)}(\mathbf{w};q,1,1)$ denote the graded Frobenius characteristic of the trivariate diagonal higher-harmonic space, with $\Dyck{r}{n}$ the set of $r$-Dyck path…

Let $m,n\in\mathbb{N}$, let $p$ divide $m$, and let $G(m,p,n)$ be the complex reflection group contained in $G(m,n)$. Its $q$-harmonic space is defined using the $q$-deformed opera…

Eventual fullness conjecture. The solution space of this linear system is all of $Alt_n^{(d)}$ for sufficiently large $d$.

Finite-dimensionality conjecture. The non-commuting Harmonic system $Har_n$ is finite dimensional for every $n$.

Let $\widehat{D}_k=\sum_{i=1}^n\partial_i^{k+1}x_i$ for $k\geq1$, and define the space of hat-harmonics by … The symmetric group $\mathfrak S_n$ acts by permuting variables, and th…