54 problems

Sendov's conjecture. The closed disk

Let $n$ point charges be placed in Euclidean space, and consider their electrostatic field or potential away from the charges. Maxwell's conjecture. The field of $n$ point charges…

Schoenberg's conjecture. The quadratic inequality

Let $\mathcal{A}$ be a hyperplane arrangement with complement $M$, let $\Phi_{\lambda}$ be its master function for weights $\lambda$, and let $\chi(M)$ denote the Euler characteris…

Let $f\colon M\to\mathbb{R}$ be a smooth function on a manifold $M$ without boundary, and let $x_0\in M$ be an isolated critical point. A cylindrical ball neighborhood is a cylindr…

Let $\mathfrak{g}$ be a simple Lie algebra, let $V_1,V_2$ be representations of $\mathfrak{g}$, and let $\Phi$ be the corresponding phase function in variables assigned to the simp…

Let $L\to M$ be an ample holomorphic line bundle over a compact Kähler manifold of complex dimension $m$, let $P(M,L)$ denote the relevant space of Hermitian metrics, and let…

Conjecture for sums of powers of quadratic polynomials. If $f\in\mathcal{QP}_n$, then $f$ has at most $2n-1$ real critical points. Moreover, if $\alpha$ is a negative integer, each…

For $|\beta|\leq 1$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the unit disk and at least one root at $\beta$. For…

Let $\beta$ satisfy $|\beta|\leq 1$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the unit disk and at least one root at…

Let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the unit disk and at least one root at $\beta$. For $P\in S(\beta)$, let…

Let $n\ge 3$ and let $f:({\mathbb C}^n,0)\to({\mathbb C},0)$ be a germ of a holomorphic function with an isolated critical point, so that $\mu(f)<\infty$. Let $d(f)$ denote the min…

Let $m$ be the complex dimension of ${\mathbb C}{\mathbb P}^m$, and let $\mathcal{N}^{\operatorname{crit}}_{N,h}({\mathbb C}{\mathbb P}^m)$ denote the expected number of critical p…

High-level universality conjecture. The high-level regime exhibits the same qualitative behavior as in the monochromatic case.

Mixed Poisson–FGF conjecture. The limiting critical point measure should exhibit mixed behavior, combining a Poisson component and an independent $\mathrm{FGF}_{1/2}(\mathbb{R}^d)$…

Uniqueness conjecture. If $\Omega$ is convex, then $\psi_\Omega$ has a unique critical point.

Let $G_p(z;\tau)$ be the Green function considered in the paper, with $p$ and $\tau$ as in the preceding results. Conjecture A. If $G_p(z;\tau)$ has $10$ critical points, then thes…

De Bruin–Sharma's conjecture. In the quartic case, the inequality

Consider the heterogeneous quadratics minimization problem … where $A_i \in {\rm Sym}(\mathbb{R}^n)$ and $Z_i$ is the $i$th column of $Z$. Critical-point count conjecture. For…

Gabrielov–Novikov–Shapiro conjecture. For all $p\geq 1$, the number of equilibria of the restriction of $V_p$ to any straight line in $\mathbb{R}^3$ is at most $2n-1$.

Let $\Omega$ be a simply connected domain, and consider the Moser-Trudinger functional … under the Dirichlet-energy constraint $\int_{\Omega}|\nabla u|^2dx=\gamma$. For sufficientl…

Generalized Borcea variance conjecture. For every $p\geq 1$ and every such choice of weights,

Let $F$ be a complex polynomial of degree $n\geq 2$ with zeros $z_1,\dots,z_n$ satisfying $|z_k|\leq 1$, and critical points $w_1,\dots,w_{n-1}$. Let $\gamma_n$ be the radius of th…

Schmeisser's conjecture. For every such choice of weights,

Let $\boldsymbol{z}$ be generic on the unit circle, let $\Phi_{m,n}(\boldsymbol{z},\boldsymbol{\xi})$ be the master function, and write $\#\operatorname{LP}(n,m)$ for the number of…