14 problems

Radicality conjecture. The ideal $I_{\Delta}$ is radical if and only if

Primality and perfection conjecture. If $\Delta$ is any simplicial complex, then

Collapsing conjecture. Suppose that

Bounded Markov complexity conjecture. If $\{1,2,\ldots,j\}$ is an independent subset of the underlying graph of $\Delta$, then the preceding bounded-format conclusion holds. This w…

Let $(S_\ell)$ be the special sequence described at the beginning of the section, and let $\mathfrak F_\ell$ be the random labelled forests associated with the coalescing partition…

Consider Dyson's classical $r$-component hierarchical model with continuous $O(r)$ symmetry, $r\geq 2$, and interaction potential $U(i,j)=-l(d(i,j))d^{-2}(i,j)$, where $d(i,j)$ is…

Zero-denominator cancellation conjecture. All numerators corresponding to zero denominators are zero.

Let $\mathbf{M}_{r,N}(\alpha,\theta_{0},\beta,\theta)$ be the number of clusters with frequency $r$ in a sample of size $N$ under the hierarchical Pitman–Yor process, and let…

Let $\Gamma$ be a pure simplicial complex whose facets all have $n$ elements. Let $\mathbf{2}$ denote the binary weight vector, and let $\operatorname{Marg}(\Gamma,\mathbf{2})$ be…

Let $\Gamma$ be a simplicial complex on $[n]$, and let $d \in \mathbb{Z}_{\geq 2}^{|\Gamma|}$ be a weight vector. Write $\operatorname{codeg}(\operatorname{Marg}(\Gamma,d))$ for th…

An anomaly-search problem may be represented by an arbitrary tree rather than a binary tree, so paths from leaves to the root can have different lengths and node distributions may…

Let $H$ be the hierarchical Anderson model with parameter $c>0$, and let $d_s$ denote its spectral dimension. Absence-of-delocalization conjecture. There is no delocalization trans…

Invariance-group conjecture. Under mild regularity conditions, the group of invariance is generated by the wreath product of this paper together with permutations of factors having…

Derrida–Hakim–Vannimenus disorder relevance conjecture. Disorder is relevant when $1/2<lpha<1$, irrelevant when $lpha<1/2$, and marginally relevant when $lpha=1/2$. More precisely,…