74 problems

Let $D$ be an optimal configuration in a planar domain, and let $\partial D$ denote its free boundary. Two-dimensional free-boundary regularity conjecture. In dimension two, the fr…

Let $\lambda$ be a singular cardinal, let $\mu<\lambda$, and let $D$ be an ultrafilter that is … -complete and $(\mu,\kappa)$-regular for every $\kappa<\lambda$. Regularity limit p…

Top-socle Lex Plus Powers conjecture. Then

Switching-rook-polynomial conjecture. The $h$-polynomial of $\mathbb{K}[\mathcal{P}]$ is equal to $\widetilde{r}_{\mathcal{P}}(t)$, and its regularity is equal to $r(\mathcal{P})$.

Let $X$ be a $W^{2,p}\times W^{1,p}$ formally Kähler manifold with complex structure $J$ and metric $g$. Low-regularity Newlander–Nirenberg conjecture. The Newlander–Nirenberg theo…

Let $p\gg 0$, and let $X$ be a $W^{2,p}\times W^{1,p}$ formally Kähler manifold with almost-complex structure $J$ and nearby form $\omega_J$. Smooth Kähler structure conjecture.…

Küronya–Pintye conjecture.

Let $b1$ be a rotation number of bounded type, and let $\Gamma_{b1}(\theta)$ denote the parametrization of the boundary of the corresponding Siegel disk. Bounded-type nonsmoothness…

Bounded-type regularity conjecture. If $b1$ is of bounded type, then the map $\Gamma_{b1}$ is at least $C^1$.

Let $K$ be an algebraically closed field, let $R=K[x_1,\dots,x_N]$, and let $P\subset (x_1,\ldots,x_N)^2$ be a homogeneous prime ideal. Write $\operatorname{reg}(P)$ for its Castel…

Let $n\geq 2$. A convex function is a function whose restriction to every line segment in its domain is convex. Write $B^n(0,1)$ for the open unit ball in $\mathbb R^n$, and let…

Assume that ${\Bbbk}$ has characteristic $2$, let $R={\Bbbk}[x,y,a,b]$, set $Q=(x^3,y^3)$ and $f=xya-(x^2+y^2)b$, and let $n=2^s$ for some $s\in\mathbb N$. Characteristic-two upper…

Let $R={\Bbbk}[x,y,a,b]$ be the polynomial ring from the preceding construction, let $S=R[z]$, and set $I=Q\cap(f,z)\subseteq S$. The source specifies that…

Let $r\geq 2$ and consider $C^r$ Anosov diffeomorphisms, with their stable and unstable distributions. A distribution is stably $C^1$ if it remains $C^1$ under every sufficiently…

Let $f$ be a $C^r$ partially hyperbolic diffeomorphism on a compact manifold $M$, with $r\geq 2$. The distribution $E^s_f$ is stably $C^1$ if, for every…

Local boundedness conjecture. Under these hypotheses, $u\in L^\infty(B_1)$.

Let $w$ be a Lipschitz subsolution to the generalized $k$-convexity equation for $\beta=1$. A measure-valued Hessian conjecture asserts that there exist signed Borel measures…

Let $\Delta$ be a simplicial tree of dimension $t-1$, let $I(\Delta)$ denote its facet ideal in the polynomial ring $R$, and let $\nu_1(\Delta)$ and $\nu(\Delta)$ denote its induce…

Fox–Sudakov's conjecture. For any graph $H$ there exists a $C_H>0$ such that for any $0<\varepsilon \le 1/2$ any $n$-vertex $H$-free graph contains an $\varepsilon$-homogeneous set…

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 $G$ be a graph, let $I(G)$ denote its edge ideal, let $I(G)^{\{s\}}$ denote the squarefree part of its $s$-th symbolic power, and let ${\rm match}(G)$ denote the matching numbe…

Smoothness conjecture. The solution $u(t)$ of the Poincaré problem is smooth.

Let $X\subseteq\mathbb{P}^r$ be a projective toric variety. Its maximal generator degree is denoted by $\operatorname{maxdeg} X$, and its regularity, degree, and codimension by…

Vorst's conjecture. If $R$ is $K_{d+1}$-regular, then $R$ is regular.

Let $G$ be a connected graph on vertex set $[n]$, let $K_m$ be the complete graph on $[m]$, let $J_{K_m,G}$ be the generalized binomial edge ideal, let $c(G)$ be the number of maxi…