159 problems

Diagonal stability conjecture. There exist positive constants $C_p>0$ such that, if $(r_k)$ is log-concave, then

Energetic stability threshold conjecture. There is $\omega_* \in (\omega_L,1)$ and $\Omega_* \in (0,1)$ such that the even wave satisfying (even-wave) is energetically stable for…

Let $E$ be an object carrying a $Z$-critical equation, and let $h$ be a metric on $E$. A strong subsolution is a metric satisfying the positivity conditions on all formal matrix de…

Let $E\to(X,\omega)$ be a Hermitian vector bundle, and let $h$ be a Hermitian metric on $E$. A strong subsolution for the $Z$-critical equation is a metric satisfying … as an…

Let $(X,H)\to(B,L)$ be a polarised fibration, meaning a fibration equipped with relatively ample polarisation $H$ and a polarisation $L$ on the base. An optimal symplectic connecti…

For $\eta>0$, consider the coupled transport–diffusion system with state variables $y$ and $\hat y$ on $[0,1]$: … with boundary conditions … Let…

Uniform boundedness conjecture. For every type $t$, there is a bounded region

Harmonic metric conjecture. There exists a harmonic metric $h$ on ${\cal B}$ if and only if ${\cal B}$ is a direct sum of $\nabla$-stable bundles. If ${\cal B}$ itself is $\nabla$-…

Suda–Tanaka–Tokushige's stability conjecture. Assume

Let $n$, $k$, $t$, and $d$ satisfy … Let $\mathcal{F}\subseteq\binom{[n]}{k}$ be $t$-intersecting. Ellis–Keller–Lifshitz stability conjecture. If … then there exists a $t$-subset…

Let $(\Gamma,\Sigma)$ be a nontrivial graph pair, meaning that $\Gamma$ and $\Sigma$ are coprime connected twin-free graphs and exactly one of them is bipartite. Assume that the gr…

Bruzzo's conjecture. The following conditions are equivalent:

LDS implies GAS of LTN equilibria. If $W-D\in\mathcal{LDS}$, then for every $u\in\mathbb{R}^n$, the linear-threshold network admits a unique globally asymptotically stable equilibr…

Let $P$ be a poset and let $M$ and $N$ be persistence modules over $P$. Let $d_I(M,N)$ denote their interleaving distance, and let $d_{DG}(M,N)$ denote the derived convolution metr…

Asymptotic semistability conjecture. There exists an integer $m_0\gg0$ such that, for every $m\ge m_0$, the syzygy bundle $S_{\mathcal{E}(m)}$ is semistable. This is proposed as a…

Let $X$ be a smooth projective variety, and let $A$ and $P$ be line bundles on $X$. Assume that $A$ is very ample, and set $L_d:=dA+P$ for every positive integer $d$. Ein–Lazarsfel…

Let $(X,\omega)$ be a compact Kähler manifold and let $D$ be a smooth divisor. Let $T$ be a maximal compact torus in the reduced automorphism group…

Bjerkevik's conjecture. On persistence modules of bounded pointwise dimension, the pruning distance is stable and Lipschitz equivalent to the interleaving distance. More precisely,…

Extension of the projected closed-loop ISS result. Under standard model-order-reduction assumptions, the closed-loop dynamics on the submanifold…

Let $u_\mathrm{pl}(\xi,\tau)$ be a modulated pulled pattern-forming front satisfying the stated pinched-double-root, periodic-wake, exponential-asymptotic, marginal-Floquet-stabili…

Let $u_\mathrm{ps}(\xi,\tau)$ be a modulated pushed pattern-forming front, namely a temporally modulated pushed front leaving a pattern-forming wake. Selection of modulated pushed…

Let $u_\mathrm{pl}$ be a rigid pulled pattern-forming front, meaning a rigid front with marginal spectral stability that leaves behind a diffusively stable periodic pattern and has…

Let $u_\mathrm{tr}$ be a pushed-to-pulled transition front. A pushed-to-pulled transition front is a front at the boundary between pushed and pulled invasion, characterized here by…

Let $\beta\in\mathbb{R}_{>1}$, let $c$ be the contraction factor associated with $A_2$, and let ${\cal M}(\theta,c,\beta)$ denote the corresponding switched-system matrix family. I…

Consider the two-dimensional semi-log canonical hypersurface singularities in the families labelled (1)--(3) in the paper. Semistability conjecture. All singularities in these thre…