26 problems

Octonionic Calabi–Yau conjecture. There exist a $C^\infty$-smooth function $u\colon M\longrightarrow\mathbb{R}$ and a constant $A>0$ such that

Full-mass conjecture. $e_n(\varphi)=1$ if and only if $u\in\mathcal{E}(\omega_0)$.

Let $Y$ be the smooth part of the limiting base of a maximal degeneration of a Calabi–Yau family, equipped with its Monge–Ampère metric and integral affine structure, and let…

Even $L_p$-Minkowski uniqueness conjecture. For $p\in[0,1)$, the Monge–Ampère equation

Schneider's conjecture. For $f_1,\ldots,f_n\in\operatorname{Conv}(\Omega)$, the mixed Hessian measure satisfies

Let $\mathcal{C}_{\theta}$ be the spherical cap, let $\alpha>\frac{1}{n+2}$ and let $\theta\in(0,\pi)$. Suppose that $h$ is a positive, strictly convex function on…

Let $u\in {\mathcal P}(B_1)$ be a plurisubharmonic function whose alternating part has finite frequencies, meaning that near the origin, in a proper complex Hopf-coordinate, it can…

Let $l,m,k$ be parameters and let $\varphi:(\mathbb R_+)^l\times \mathbb R^{m}\to \mathbb R_{\geq 0}$ solve … in $(\mathbb R_+)^l\times \mathbb R^{m}$, with … on…

Let $Y_{[3]}^\Gamma$ be a resolved threefold with exceptional divisor $\mathcal{ED}$, and let the Ricci-flat metric and the Kähler metric on $\mathcal{ED}$ be those considered in t…

Higher-dimensional decomposition conjecture. For every $D\in\mathcal{C}^{m,\gamma}(\bar\omega,\mathbb{R}^{d\times d}_{\mathrm{sym}})$,

Let $D=B_1$ be the unit ball in ${\mathbb C}^n$, and let $u$ be a plurisubharmonic function on $B_1$ that is locally bounded outside the origin. The Monge–Ampère measure…

Rotational symmetry conjecture. Every such solution $u$ on $\mathbb{R}^n$ is rotationally symmetric.

Let $\varphi$ solve the complex Monge–Ampère equation considered in the paper, with right-hand side $F$. The question concerns estimates for the regularity of $\varphi$ under assum…

Let $X_{NA}$ be the non-Archimedean analytification, let $\mathcal{E}^{1}(X_{NA})$ be the finite-energy space, let $\mathcal{H}(X_{NA})$ be the space of non-Archimedean model poten…

Alesker–Verbitsky conjecture. For every such $f$, the quaternionic Monge–Ampère equation admits a unique smooth solution $\phi$, up to addition of a constant.

Metric SYZ conjecture. As $t\to0$, $(X_t,\widetilde{\omega}_t)$ converges in the Gromov–Hausdorff topology to a compact metric space $(Z,d)$ with an open dense subset $Z_0$ such th…

Let $(X,L)$ be a polarized manifold, let $\mathcal{E}^{1,\rm NA}(L)$ be the finite-energy non-Archimedean metrics, let ${\mathcal H}^{\rm NA}(L)$ be the smooth positive non-Archime…

Schwarz symmetrization conjecture. One has

Trudinger–Wang's conjecture. The first boundary value problem has a unique smooth solution $u$ if $\phi$ is affine mean convex. This conjecture concerns existence and uniqueness fo…

Trudinger–Wang's conjecture. If $n\leq 9$, then $u$ is an elliptic quadratic polynomial. If $n\geq 10$, then there exists a smooth, locally uniformly convex solution of the equatio…

Let $u\colon\mathbb{R}^2\to\mathbb{R}$ be smooth and locally uniformly convex, and suppose its graph is an affine maximal graph, meaning that $u$ satisfies the affine maximal surfa…

Tosatti–Weinkove's conjecture. There exists a unique pair $(u,b)$, with $u$ a smooth function on $M$ and $b$ a constant, such that

Calabi–Yau threefold symplectic Monge–Ampère conjecture. The equation has a unique solution $\varphi:M\to\mathbb{R}$, modulo adding a constant, such that…

Let $S_T=[0,T]\times\mathbb{R}$ and let $\varphi_\infty$ be the quantum analytic continuation potential for the Cauchy data $(\omega_{\varphi_0},\dot{\varphi}_0)$. Define…

Quaternionic Calabi conjecture. If the necessary condition