866 problems

A domain in the Riemann sphere $\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ is a connected open set. A circle domain is a domain whose complementary components are points or close…

Sendov's conjecture. The closed disk

Let $D$ be a strongly pseudoconvex domain with smooth boundary. Let the Cheng–Yau metric be the canonical complete Kähler–Einstein metric on $D$, and let the Bergman metric be the…

Let $\Omega\subset\mathbb C^n$ be a smoothly bounded pseudoconvex domain with a non-compact automorphism group, and let $p\in\partial\Omega$ be a boundary orbit accumulation point,…

Let $\Omega$ be an open Riemann surface admitting a nontrivial Green function $G_{\Omega}$. For $z_0\in\Omega$, let $c_{\beta}(z_0)$ denote its logarithmic capacity and let…

Let $H_{n,m}(z)=p(z)+\overline{q(z)}$ be a harmonic polynomial with $\deg p=n$ and $\deg q=m<n$. The Wilmshurst refinement. For each $n,m$ with $m<n$, the maximum number of zeros o…

Let $D\subset\mathbb{C}^n$ be a bounded pseudoconvex domain. The Bergman metric of $D$ is the canonical Kähler metric associated with its Bergman kernel. A domain is biholomorphic…

Let $\mu\in\mathcal P(\mathbb C)$ be a probability measure on $\mathbb C$, let $(X_i)_{i\in\mathbb N}$ be independent and identically distributed random variables with law $\mu$, a…

Let $\phi$ be a germ of a plurisubharmonic function at a point $x$ on a complex manifold, and let $c_x(\phi)$ denote its complex singularity exponent. Demailly–Kollár openness conj…

Krzyż's conjecture. For every $f\in\Omega$ and every $n\geq 1$,

Let $p(x)$ be a real polynomial of even degree $n$. For a real polynomial $q$, write $\sharp_r q$ for the number of its real zeros. Shapiro's Conjecture 12. … The conjecture concer…

Let $P$ be a complex polynomial of degree $n\ge2$ satisfying $P(0)=0$ and $P'(0)=1$. A point $\zeta$ is a critical point of $P$ when $P'(\zeta)=0$. Smale's mean value conjecture. T…

Analytic version of Jonsson–Mustaţă's conjecture. If $c_o^{\mathfrak{q}}(\varphi)<+\infty$, then

An open connected Riemann surface is a connected one-dimensional Stein manifold. Forster's conjecture. Every open connected Riemann surface embeds properly holomorphically into…

A circle domain $\Omega\subset\widehat{\mathbb C}$ is a domain whose boundary components are points or circles. It is conformally rigid if every conformal map of…

Let $n \geq 1$, and let $p\colon\mathbb{C}\to\mathbb{C}$ be a monic polynomial of degree $n$. For $R\geq 0$, define … and … The Erdős–Herzog–Piranian conjecture. If…

Mocanu's conjecture. Then the harmonic function $f=h+\overline{g}$ is univalent in $\mathbb{D}$. This conjecture concerns a sufficient condition for univalence of a sense-preservin…

Consider the meromorphic Cauchy transform … where $|t_n|\to\infty$ and the series is understood under the usual local convergence condition. Clunie–Eremenko–Rossi conjecture. If…

Let $y(\lambda)$ be the tritronquée solution of Painlevé I, and let $\alpha\in\mathbb{C}$ be a pole of this solution. Dubrovin's conjecture. If $\alpha$ is a pole of the tritronqué…

Let $A_n(\alpha,\beta,\omega)$ be defined by the Maclaurin expansion … where $|\omega|=1$ and $\alpha,\beta\in(0,1]$. Brannan's conjecture. For every positive odd integer $n$, … Th…

D'Angelo's degree conjecture. The degree bound can be taken to be

Schoenberg's conjecture. The quadratic inequality

Bank–Laine conjecture. If $\rho(A)\notin\mathbb{N}$, then

Let $f$ be an analytic function, and consider its graph and pluripolar hull. The Levenberg–Martin–Poletsky conjecture. The pluripolar hull of the graph of $f$ is equal to the graph…

Bombieri's odd-even refinement. If