65 problems

Let $P_{abc}$ be a unital symmetric real tensor, meaning that its indices are symmetric and its unit conditions are $P_{1ab}=P_{a1b}=P_{ab1}=\delta_{ab}$. A positive tensor rank de…

For every integer $p\geq 1$, the resonance transformations of the $(2,2p+1)$ minimal string coincide with the transformations induced by interchanging the two coordinates,…

For every bounded simply connected planar domain $\Omega$ with infinite-mass boundary condition, every mass $m\ge 0$, and the disk $B_{|\Omega|}$ having the same area as $\Omega$,…

Let $G$ be a connected, locally finite, infinite network, and let $\mathfrak{F}_{\mathrm{F}}$ denote its free uniform spanning forest. Is it always true that…

For the relevant semiclassical magnetic Schrödinger inverse problems, the spectral data should determine the coefficients in the following symmetry-restricted settings: (i) an even…

For the Fermi–Pasta–Ulam system in the standard continuum-limit scaling, determine whether solutions with initial data at $L^2$ regularity converge, in the appropriate norms, on ev…

For every smooth magnetohydrostatic equilibrium $(\Omega,p,B)$ in Euclidean $3$-space, where $\Omega$ is an embedded solid torus, the regular level sets of the pressure $p$ are nes…

For the three-dimensional incompressible Navier–Stokes equations $\partial_t u+(u\cdot\nabla)u=-\nabla p+\nu\Delta u$, $\nabla\cdot u=0$, with $\nu>0$ and smooth divergence-free in…

For every consistent theory of quantum gravity with spacetime dimension $d$, tangential structure $\xi$, and background data encoded by a space $X$, the associated bordism group is…

For every speed $c\in(0,\sqrt{2})$, there exists a nonconstant finite-energy profile $\psi:\mathbb{R}^2\to\mathbb{C}$ such that the traveling wave $\Psi(t,x)=\psi(x-ct e_1)$ solves…

Let $H_V=-\frac{d^2}{dx^2}+V$ on $L^2(\mathbb{R})$, where $V\colon\mathbb{R}\to\mathbb{R}$ and $V\in L^{3/2}(\mathbb{R})$. If the negative eigenvalues of $H_V$ are written as…

Given two infrared-renormalized van Hove–Miyatake models, determine necessary and sufficient conditions for them to be unitarily equivalent; that is, characterize when there exists…

Let $f_t$ be a sufficiently regular solution of the spatially homogeneous Landau equation with Coulomb interaction, and let…

For every integer $N\ge 1$, every collection of distinct points $\mathbf a_1,\ldots,\mathbf a_N\in\mathbb R^3$, and every collection of nonzero charges $q_1,\ldots,q_N$ having the…

Every gapped ground state satisfies approximate Haag duality.

For every smooth projective effective orbifold curve $X$, the total descendant Gromov–Witten partition function $\mathcal{Z}_X$ satisfies the Jiang–Tseng Virasoro constraints…

For every integer $n\ge 4$ and every choice of positive masses $m_1,\ldots,m_n$, consider the Newtonian $n$-body problem in $\mathbb{R}^3$. Let $\mathcal{S}_n$ be the set of collis…

For generic coupling $\beta=-\epsilon_1/\epsilon_2$, the four-point conformal block for the $SU(2)$ theory with four fundamental hypermultiplets equals, coefficientwise at every in…

For every Böhm Einstein metric $g$ on a manifold of the form $S^{k+1}\times S^l$ or $S^{k+l+1}$, the Lichnerowicz Laplacian on transverse-traceless symmetric $2$-tensors has a nega…

For the symmetric ACRF-4 registration model, determine whether the canonical permutation-symmetry decomposition $\mathbb{R}^{4}=\operatorname{span}\{\mathbf{1}\}\oplus H_{4}$, wher…

For every integer $n\ge 2$ and every admissible marked configuration of $n$ complete pairwise disjoint timelike affine worldlines in Minkowski space, let $F_1,\ldots,F_n$ be the as…

For every black-hole spacetime with Hawking temperature $T_H$, the least-damped quasinormal mode with frequency $\omega_0$ satisfies…

Determine the leading asymptotic of the largest eigenvalue of the $N$-Majorana quartic SYK Hamiltonian as $N\to\infty$.

Let $(a_n)_{n\ge 1}$ be the Matryoshka numbers of Definition 1.1 in Liu's note. Then there exists a real number $S$ with $0<S<\infty$ such that … … and … Moreover, …

Conjecture 1.3 (Maxwell; [11], see also Section 4). The total number of points of equilibrium, all assumed nondegenerate, of any configuration with $l$ charges in $\mathbb{R}^{3}$…