426 problems

Let $\mathbb{R}\langle \xi\rangle$ be the real closed field of algebraic Puiseux series, let $\mathcal{P}=\{P_1,\ldots,P_s\}\subset \mathbb{R}[X_1,\ldots,X_n]$, and set … For a bou…

Setting. For a Navier-Stokes velocity field with vorticity $\omega$, direction $\xi = \omega/|\omega|$, and peak set $P(t)$ (the high-vorticity region), define the scale-invariant…

Let $n$ be a positive integer, and let an ordinary double point denote the corresponding $n$-dimensional klt quadratic singularity. Ordinary double point volume gap conjecture. The…

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McKernan's conjecture. Fix a positive integer $r$ and a real number $0<\epsilon\leq 1$. There exists $\delta>0$ depending only on $r,\epsilon$ such that, whenever…

Nonnegativity conjecture. All stringy Hodge numbers $h^{p,q}_{\rm st}(X)$ are nonnegative.

Let asymptotically flat initial data for Einstein's field equations be given, and let its maximal development be the corresponding spacetime. A future null infinity $\mathcal{I}^+$…

Calabi–Yau type conjecture. Then $X$ is of Calabi–Yau type.

Let $W$ be an invertible polynomial in $n$ variables, let $R=\mathbb{C}[x_1,\dots,x_n]/(W)$, and let $W^\ast$ be its transpose polynomial. Write $D^b_{\mathrm{sing}}(R)$ for the bo…

Let $(\mathcal{M},g)$ be a $3+1$-dimensional Lorentzian spacetime solving the Einstein field equations, arising as the maximal future development of generic, regular, asymptoticall…

Let $X$ be a singular variety and let … be a resolution of rational singularities. Write $\textbf{D}^{b}(X)$ and $\textbf{D}^{b}(\widetilde{X})$ for their bounded derived categorie…

Let $p$ be a prime number and define … Let $E_n$ denote the Euler number, equivalently the number of linear extensions of the zigzag poset of size $n$. Watanabe–Yoshida conjecture.…

Let $V$ be a normal affine variety of complex dimension $n$ with an isolated singularity at $0$ that is numerically $\mathbb{Q}$-Gorenstein. Write $\operatorname{min.disc}(V,0)$ fo…

Bridgeland stability conjecture. The category $\operatorname{\mathcal{K}\mkern-1mu\mathit{u}}(Y)$ admits a Bridgeland stability condition.

Consider the mean curvature flow of embedded hypersurfaces, and let a blowup limit mean a limit obtained by parabolic rescaling near a singularity, counted with its varifold multip…

Lower semi-continuity conjecture. The function $x\mapsto \operatorname{mld}_x(X,\mathfrak{a})$ is lower semi-continuous.

Let … be a $\mathbb{Q}$-Gorenstein flat family of klt singularities with a section $t\in T\mapsto x_t\in\mathcal{X}_t$. Constructibility conjecture. The function … is constructible…

Let $P\in X$ be the germ of a smooth threefold, let $\mathfrak{m}$ be the maximal ideal in $\mathscr{O}_X$ defining $P$, and let … be $\mathbf{R}$-ideals on $X$. Fix a finite subse…

Let $M_t$ be a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$, with $n dleq 6$, and suppose the flow develops a first finite singular time. Mean-curvature blow-up…

Jonsson–Mustață's equivalent conjecture. There is a quasimonomial valuation $\nu$ on $\mathcal{O}_o$ that computes $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$, namely

Shokurov's conjecture. Fix a positive integer $r$ and a real number $0<\epsilon\leq 1$. There exists $\delta>0$ depending only on $r,\epsilon$ such that, if $\dim X-\dim Z=r$,…

Shokurov's index conjecture. There exists a positive integer $r(n,a)$ such that, whenever $\operatorname{mld}_p(X)=a$, the Cartier index of $K_X$ at $p$ is at most $r(n,a)$.

Let $(X,\Delta+M)/Z$ be a generalized pair. Define … where $|X|_{\mathrm{cl}}$ denotes the set of all closed points of $X$ with the Zariski topology. Lower semi-continuity conjectu…

Canonical-threshold ACC conjecture. The set $\mathcal{T}_{n}^{\textup{can}}$ satisfies the ascending chain condition.

Let $P\in X$ be the germ of a normal variety, and let $S+B$ be an effective $\mathbf{R}$-divisor on $X$ such that $S$ is a normal prime divisor not appearing in $B$. Assume that…