186 problems

Let $G$ be a graph with edge ideal $I(G)$, and let $I(G)^{(t)}$ denote its $t$-th symbolic power. The Castelnuovo–Mumford regularity of an ideal is denoted by $\operatorname{reg}$.…

Let $F$ be a field, and let $X$ be an affine finite-type $F$-scheme of dimension $d$. A scheme is regular if all of its local rings are regular, and it is $K_m$-regular when the ca…

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 $r(\mathcal{P})$.

Küronya–Pintye conjecture.

Marmi–Moussa–Yoccoz conjecture. The function $\Upsilon$ is $(1/2)$-Hölder continuous. This conjecture concerns the regularity of the correction to the Bryuno asymptotic for the con…

Let $G$ be a graph, let $I(G)$ be its edge ideal, and let ${\rm reg}(I(G))$ denote its Castelnuovo–Mumford regularity. For each integer $s\geq 1$, Alilooee–Banerjee–Beyarslan–Hà's…

Let $G$ be a graph on $n$ vertices, let $J_G$ be its binomial edge ideal in a polynomial ring $S$, and let $h_{S/J_G}(t)$ be the numerator polynomial of the Hilbert series of…

Fano-type conjecture. Globally $+$-regular schemes are of Fano type.

The clique-count regularity conjecture. For every graph $G$,

Let $f$ be the $1$-periodic continuous function … A function is Hölder of exponent $1/2$ if there is a constant $C>0$ such that … for all $\theta_1,\theta_2$. Marmi-Moussa-Yoccoz c…

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…

De Giorgi's conjecture. Any initial $n$-dimensional smooth submanifold $\varphi:M\to\mathbb{R}^{n+m}$ evolves by the gradient flow of $\mathcal{G}_k$ without developing singulariti…

Let $u:\Sigma\times[0,T)\to N$ be a solution of harmonic map flow in the critical dimension $\dim(\Sigma)=2$, and define its body map by … Assume that the target $N$ is real-analyt…

Banerjee–Beyarslan–Hà conjecture.

Let $\rho_t$ denote a super-Brownian motion and let $L_x$ be its total occupation-time density, defined by … for suitable test functions $\phi$, in dimensions $d\leq 3$. The Hölder…

Let $A$ be a polynomial ring over a field $k$. Let ${\mathfrak m}$ be the maximal graded ideal of $A$, and let $I$ be an equigenerated homogeneous ${\mathfrak m}$-primary ideal. Wr…

Let $N$ be a smooth Riemannian manifold, let $g\in C^{1,\alpha}$, and let $E\subset N$ be a Caccioppoli set satisfying the hypotheses of Corollary 2 with the stability assumption (…

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

Chen's regularity conjecture. Any $C^{1,1}$ function which attains the absolute minimum of the K-energy or the modified K-energy is actually smooth.

Let $H:\operatorname{\mathbb{R}}\times\operatorname{\mathbb{R}}_+\to\operatorname{\mathbb{R}}$ be a Borel function, and let $N$ be a continuous local martingale with…

Regularity-jump conjecture. We conjecture that

Let $I$ be a reduced equidimensional homogeneous ideal, and let $m(I)$ denote its regularity. In the preceding bound, part (a) asserts the linear estimate $m(I)\leq \deg Z-n+r+1$ u…

Let $Z\subset \mathbb{P}^n$ be a reduced subscheme purely of dimension $r$, and let $I=I(Z)$ be the full homogeneous ideal of functions vanishing on $Z$. If $Z$ is irreducible, the…