20 problems

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…

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…

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…

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 $2n-1$.

Maxwell's conjecture. The electrostatic potential defined by $n$ point charges in $\mathbb{R}^3$ has at most $(n-1)^2$ equilibria.

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 $z_1,\dots,z_n$ and critical points $w_1,\dots,w_{n-1}$. Schoenberg's conjecture. … with equality if and only if all…

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…

Let $\boldsymbol{z}$ be a generic configuration on the unit circle, and let $\Phi_{m,n}(\boldsymbol{z},\boldsymbol{\xi})$ be the master function. Involution-symmetry conjecture. Ev…

Consider linear neural networks trained on $m$ data points. The results referred to as Proposition … , Corollary … are observed computationally to hold when $m=2$. Extension conjec…

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…

Magnanini's conjecture. The number of critical points satisfies

Circular-domain critical-point conjecture. For $k\geq 3$,

Quadrature-domain critical-point conjecture. The number of critical points satisfies

Extremal critical-point conjecture. For every $n\geq 2$, and for each $k=1,2,\ldots,n+1$, such a domain exists. The bound is attained for $k=n+1$ by the extremal construction descr…

For each zero $Z_{k,n}$ of a random polynomial $p_n$, let $\zeta_{k,n}$ be a nearest critical point, and define … … Here $\mathcal N_{\mathbb C}(0,1)$ denotes the standard normal d…