42 problems

Let $S$ be a polynomial ring over a field, and let $I_1,\ldots,I_d$ be ideals in $S$ generated by linear forms. The regularity of a homogeneous ideal $I$ is its Castelnuovo–Mumford…

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

Let $\mathfrak g(\infty)$ be a locally simple Lie algebra, and let $\operatorname{U}(\mathfrak g(\infty))$ be its universal enveloping algebra. An augmentation ideal is an ideal ar…

Buchsbaum–Eisenbud conjecture. Every finite free resolution is a free resolution of a three-generated ideal.

Let ${\cal I}$ and ${\cal J}$ be two of the ideals considered in the paper such that ${\cal I}\not\subseteq {\cal J}$. Covering-number separation conjecture. It is consistent that…

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…

Let $R$ be the graded or local two-dimensional regular ring considered in the paper, let $I$ be a contracted ideal, let $h_i(I)$ denote the coefficients of the $h$-vector of $I$, a…

Postnikov–Shapiro minimal-resolution conjecture. For $n=3$ and $\phi$ linear, the displayed complex is the minimal free resolution of $J_\phi$. The conjecture predicts that the min…

Postnikov–Shapiro Hilbert-series conjecture. For every linear degree function $\phi$, the Hilbert series satisfy

The paper discusses definable ideals $I$ whose associated forcing $P_I$ is proper but may fail to be $<\omega_1$-proper, or whose properness may fail to be absolute throughout forc…

Let $\mathcal I$ be a tall $F_\sigma$ ideal, and let $c_{0,\mathcal I}$ denote the corresponding ideal-defined Banach space. Natural isomorphism conjecture. All tall $F_\sigma$ ide…

Dualization conjecture.

Let $P$ be a bqo, and consider bqo-preserving operators such as $P\mapsto P^n$ for fixed $n$ and $P\mapsto T(P)$, where $T(P)$ is the Kruskal ordering. The structure of the iterate…

Let $\textsf{non}^*(\mathcal{ED}_{\textsf{fin}})$ denote the corresponding cardinal invariant for the ideal of eventually different functions, and let…

Let $I$ be an ideal of $\mathcal O_n$. Say that $I$ is diagonal with respect to coordinates $x_1,\dots,x_n$ if it is generated, up to integral closure, by pure powers of these coor…

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…

Let $I$ be the three-generated ideal family referred to in the cited conjecture, and let the degrees of its three generators vary. The author's super-polynomial regularity conjectu…

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

Let $P_T$ be the Padé matrix defining the Taylor variety $\mathcal{T}^n_{d,e,m}$, let $\hat P_T$ be the matrix obtained from $P_T$ by deleting its leftmost column, and let…

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 $f\in\mathcal{I}$.

Modified limit-ordinal conjecture. For every ideal $\mathcal{I}$, the following conditions are equivalent: