17 problems

Let $I$ be a $c3$-ideal on $2^\omega$ or $\omega^\omega$, and let $P_I$ be the corresponding factor poset. Assume that $P_I$ is proper and has continuous reading of names. Trace-id…

Dualization conjecture.

Let $(R,\mathfrak{m})$ be the local ring under consideration, let $I\subset R$ be an ideal, let $\mathcal{R}(\mathfrak{m})=\bigoplus_{k\ge0}\mathfrak{m}^k$ be the Rees algebra of…

Threshold-equality conjecture. There are infinitely many primes $p$ for which

Let $\beta=\bigwedge_{n_0<\omega}\bigvee_{n_1<\omega}\cdots\omega^{k-1}\times A_{(n_0,n_1,\ldots,n_{k-1})}$ be the indicated iterated operation on a family of sets…

The many-variable containment conjecture. For any $r\geq 2$ and $n\geq 1$,

Let $N$ be a positive integer, and let $\mathfrak{J}(N)=J(N)\cap k[x,y]$ be the contracted ideal. The contracted ideal monomiality conjecture. … The preceding theorem proves contai…

Defining-ideal conjecture. The Zariski closure of $SOT_{2,n}$ in $\mathbb{C}^{n^4}$ is cut out by the vanishing of $\mathcal P_n$, $\mathcal Q_n$, and $h_n$. The ideal defined by t…

Full numerator criterion. For every $f\in\mathbb{C}[z_1,z_2]$, $f/p$ is locally $H^{\infty}$ if and only if $f\in\mathcal{I}$.

Debs–Saint Raymond conjecture. For every analytic ideal $\mathcal{I}$ and $0<\alpha<\omega_1$,

Let $\kappa=\nu^+$, and let $\theta$ be a regular infinite cardinal less than $\kappa$ with $\theta\ne\operatorname{cf}(\nu)$. Let $NS_\kappa\vert E^\kappa_\theta$ denote the restr…

Let $\mathcal{I}$ be a fragmented ideal. The associated Rothberger number $\mathfrak{b}(\mathcal{I})$ is the least size of a family witnessing the corresponding Rothberger property…

Topological extension conjecture. The functors $\widetilde{\mathscr{T}}$ and $\mathcal{I}$ are naturally isomorphic.

Invariant partial ideal conjecture. A partial ideal of a $C^*$-algebra arises from a total ideal if and only if it is an invariant partial ideal. Consequently, the map…

Let $\mathcal{A}$ be a unital $C^*$-algebra. A partial ideal is a choice of a closed ideal $I_V$ in each unital commutative sub-$C^*$-algebra $V\subset\mathcal{A}$ such that, whene…

Representation conjecture. If $J$ is a dense $\mathbf{F}_{\sigma\delta}$ weakly selective ideal on $\omega$, then there exists a Polish space with a countable base $\mathcal{O}$ an…

Let $\mathcal{I}$ be an ideal defining a subscheme, let $\pi$ be an embedded resolution of $\mathcal{I}$, and let $Z_{\mathrm{top},\mathcal{I}}^{(d)}$ denote the associated topolog…