70 problems

Upper-bound conjecture. The classical Hamiltonian satisfies the bound

De Klerk–Pasechnik conjecture. For every graph $G$,

Let $H_n=[0,1]^n$ be the hypercube, described by $g_i=x_i-x_i^2\geq 0$ for $i=1,\ldots,n$. For even $n$, let $M_{n,n}(\mathbf g)$ be the degree-$n$ truncated quadratic module gener…

Let a biquadratic form be a homogeneous polynomial of degree two in each of two variable vectors, and let its SOS rank be the smallest number of squares of bilinear forms in a sum-…

Let $V$ be the variety under consideration, let its RR discriminant be defined by a polynomial, and call a factor isotropic when it belongs to the isotropic part of that discrimina…

Collins–Dykema–Torres-Ayala conjecture. This coefficient is non-negative.

Sun's four-square conjecture. Every such $n$ can be expressed as

Sun's conjectures. If $n\equiv10\pmod{24}$, then

Let $obreakforall obreaktext{?}$ For every $obreaktext{?}$

Let $m,n\geq 2$. Denote by $BSR(m,n)$ the maximum sum-of-squares rank of an $m\times n$ biquadratic form, and let $z(m,n)$ be the maximum number of edges in a bipartite graph with…

Let $n=2$ and let $T$ be any tensor defining the quartic-regularized model $m_3$. A necessary and sufficient global optimality condition with no relaxation gap should hold for ever…

Let $P(\mathbf{x},\mathbf{y})$ be a $3 \times 3$ symmetric PSD biquadratic form, so $P(\mathbf{x},\mathbf{y})\geq 0$ for all $\mathbf{x},\mathbf{y}\in\mathbb{R}^3$. A bilinear form…

Let $\mathcal{A}$ be a symmetric biquadratic M-tensor: a symmetric biquadratic tensor that can be written as $\mathcal{A}=\alpha\mathcal{I}-\mathcal{B}$, where $\mathcal{I}$ is the…

Let $X_0,X_1,\ldots,X_r$ be vector fields defining a sum-of-squares operator, and let the associated control system be the system denoted by … is sufficient for global hypoelliptic…

In the symmetric Tensor Principal Component Analysis model, let $x\in\{\pm1\}^n$ be an unknown signal and observe, for $i_1<i_2<\cdots<i_r$, measurements … where the noise variable…

de Klerk's conjecture. For every graph $G$,

For $k\geq 1$, let $G_k$ be the graph obtained from the complete bipartite graph $K_{k+1,k+1}$ with bipartition $U=\{u_0,\ldots,u_k\}$ and $V=\{v_0,\ldots,v_k\}$ by subdividing eac…

Let $M_{Dir}$ be the Dirichlet-polynomial matrix. Suppose $1<\alpha<2$ and $T=N^\alpha$. Let $W$ be a large-value set at threshold $N^\sigma$. Dirichlet sum-of-squares barrier conj…

Fix $1<\alpha<2$, set $T=N^\alpha$, and let $M_{ran}$ be a random $T\times N$ Gaussian matrix. Let $v\in\mathbb R^N$ satisfy $\|v\|_\infty\leq1$, and suppose…

Let $S$ be a multiset of samples, let $\boldsymbol{\theta}>0$ and $\boldsymbol{\rho}e 0$, and let $m otin e 0$? A sample set is $(epsilon,delta,m)$-certifiably resilient if there i…

Let $X$ and $Y$ be arbitrary real Toeplitz matrices, let $\langle X,Y\rangle$ denote their Frobenius inner product, and let $\|\cdot\|$ denote the Frobenius norm. Lu–Wenzel conject…

Let $\Gamma_{n,d}$ be the class of symmetric degree-$d$ polynomials in $n$ variables. For a symmetric hyperbolic polynomial $p$, let its associated operator be the operator defined…

Let $\Gamma_{n,d}$ denote the class of symmetric degree-$d$ polynomials in $n$ variables, and let $\Delta_{\vec{1},\vec{1}}p$ be the indicated second directional derivative. A poly…

The paper studies two kinds of computational evidence for dense subgraph detection: failure of the sum-of-squares (SoS) hierarchy and failure of low-degree polynomial tests. SoS–lo…

Extension conjecture. A version of the paper's main theorem is true for ternary quartics and matrix polynomials.