477 problems

Tian's properness conjecture. A cscK metric exists if and only if the Mabuchi functional is coercive modulo $G$.

Collins–Jacob–Yau conjecture. The dHYM equation admits a solution if and only if, for every proper subvariety $V\subset X$,

Let $g_{a,M}$ denote a member of the two-parameter family of Kerr black-hole spacetimes, with zero cosmological constant and $a<M$. Kerr stability conjecture. Any slightly perturbe…

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…

Suda–Tanaka–Tokushige's stability conjecture. Assume

Let $A^*$ be the interconnection-strength parameter, let $d$ be the transmission delay, and let the values in Table be the numerically identified critical delays at which purely im…

Let an elliptic equilibrium point be an equilibrium of a Hamiltonian system, with quadratic part of the Hamiltonian function at the equilibrium point. The quadratic part is sign-de…

Stability conjecture. When these stability-type cohomological obstructions vanish, the deformed Hermitian–Yang–Mills equation admits a solution.

CSC metric–polystability conjecture. The projective bundle $P(E)$ admits a CSC Kähler metric if and only if $E$ is polystable.

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

In the $N$-body problem, let $N\geq 3$. A mass is dominant when it is much larger than the other masses. Moeckel's conjecture. There is no possibility of stable motions unless ther…

Stable transduction conjecture. A hereditary class of graphs is stable if and only if it is a transduction of a nowhere dense class of graphs.

Stability conjecture. The stability of the quasiperiodic invariant circle and the stability of periodic solutions with $P\approx m/n$ should be the same.

Let $Z>0$ and consider the families of traveling-wave solutions of model C bifurcating from the stationary solution at solutions $P_0$ of the bifurcation condition … ordered by inc…

Let $f_r(n,t)$ be the smallest number such that every $K_{r+1}$-free graph $G$ with at least $e(T_r(n))-t$ edges can be made $r$-partite by deleting $f_r(n,t)$ edges. A pentagonal…

Figalli–Glaudo's nonlinear stability conjecture. For $\nu\geq 2$,

Normal-index-one conjecture. The normal index of $\phi$ is always equal to one:

Let $\mathcal{F}$ be a $k$-graph on vertex set $[n]$. Let $\nu(\mathcal{F})$ denote its matching number and let $\tau(\mathcal{F})$ denote the minimum size of a vertex cover, meani…

Let $X$ be a compact Kähler manifold with Kähler form $\omega$, let $[\alpha]$ be a real $(1,1)$-cohomology class, and for every irreducible analytic subvariety $V\subseteq X$ defi…

High-noise uniqueness and stability conjecture. In the high-noise case, there is at most one stationary state which is homogeneous in time and globally asymptotically stable.

Collins–Yau's conjecture. The following are equivalent:

Fix a network with queue $m$, tail-exponent set $\Gamma$, and the notions of RDS and special $\epsilon$-Lyapunov function as in the preceding corollary. Converse Lyapunov conjectur…

Consider the proposed dispatching policy for a system with renewal arrivals and generally distributed service times. Stability conjecture. The policy is always stable, even with re…

Odd-cycle blowup conjecture. There is a $C_{2k+1}$-blowup $G^*$ such that

Consider an extended nonlinear program with local minimizers and associated Lagrange multipliers. A local minimizer is fully stable when it has the stability property defined in th…