25 problems

Let $M=\left\{b;\left(\epsilon,g,(t,k)\right);\left(\ \mid \ \right); \left((p_1,q_1),\ldots,(p_r,q_r)\right)\right\}$ be a non-orientable closed irreducible and $\mathbb P^2$-irre…

Consider a discrete-time regulatory network and let $C(t)$ denote its complexity at time $t$. Polynomial complexity and finite asymptotics conjecture. The complexity is always boun…

Let $c(M)$ denote the complexity and ${\rm Vol}(M)$ the volume of a hyperbolic 3-manifold. Fix a topological boundary $\Sigma$, and let ${\mathcal{M}}_{\Sigma}$ be the set of hyper…

Let $t$ be a positive integer, and let $G$ be a graph of simple treewidth at most $t$. For integers $r,k\geq 0$, write $\operatorname{pc}_r(G,k)$ for the maximum, over all vertex s…

Let $H_N$ be a mixed $p$-spin Hamiltonian on the sphere, let $\mathcal{N}(D)$ denote the number of its local maxima on $S_{N-1}$ whose normalized radial derivative lies in $D$, and…

Let $X\rightarrow Z$ be a Fano type morphism, and let $(X,B)$ be a log Calabi–Yau structure over $Z$, meaning that $(X,B)$ is log canonical and $K_X+B\sim_{\mathbb{Q},Z}0$. Its rel…

Periodicity conjecture. For every $k\geq 3$, the function $b_{\mathbf{t}_{m},k}(n)$ is ultimately periodic with period $m^{k}$.

Let $X$ be the minimal dynamical system generated by the substreetution tree ${\mathfrak J}$. For $\mathfrak{A}\in X$, define … Let $R_n$ be the number of patches of size $n$ appea…

Let $X$ be the minimal dynamical system generated by the fixed substreetution tree ${\mathfrak J}$, and let $p(n,\mathfrak{A})$ denote the number of $n$-step preimages of…

The AIDAL method is analyzed under assumption (A3) and the finiteness condition $D_h<\infty$, where $D_h$ denotes the relevant Bregman-diameter quantity, and achieves an iteration…

Local-minima determinant conjecture. For $\xi(x)\in\mathcal D$,

Determinant self-averaging conjecture. To leading order as $N\to+\infty$, at fixed $L\geq 1$ and $0\leq d\leq 4$, one may replace

Polynomial-time solvability conjecture. The problem of deciding whether

Initial-state conjecture. Finding an initial state for a general partition adjacency matrix instance is NP-hard.

Let $A$ be a semialgebraic set inside the unit cube $I^n\subset\mathbb R^n$, and let $D(A)$ denote its diagram. Let $C(\epsilon,A)$ be the analytic approximation complexity at accu…

Homogeneous-demand conjecture. To reach $J_{max}$ around $\rho=0.25$ and $\rho=0.75$, it is necessary to have a homogeneous demand on all streets, so as to prevent “idling” of some…

Lens-space complexity equality conjecture. For every $p\ge 3$,

Let $\mu$ be an SRB measure for a $C^{k}$ diffeomorphism of a compact surface with Lyapunov exponents $\lambda_{-}<0<\lambda_{+}$, and let $d_{\mu}$ denote its dimension. Suppose t…

A cograph is a graph obtained from a single vertex by repeatedly applying disjoint union and complementation. For a graph $G$, the edge-clique cover problem asks whether the edges…

Let $M$ be a closed connected $3$-manifold. The Matveev complexity is denoted by $c(M)$, the modified Heegaard complexity by $c_{HM}(M)$, the non-minimal generalized GM-complexity…

Let $w$ and $w'$ be positive braid words of length at most $\ell$, with no fixed bound on the indices of their generators, and let $\operatorname{compl}_{\curvearrowright}(w,w')$ b…

Let $c$ be a natural number, and let a complexity-$c$ irreducible 4-manifold mean an irreducible 4-manifold having complexity $c$. Finiteness conjecture. For every natural number…

Quadratic upper-bound conjecture. For all $n$-expressions $u,v$ of length $\ell$, the reversing complexity

Complexity-grouping conjecture. This grouping of the models is sufficient to take into account the richness of $\mathcal{M}_n$ for the optimal calibration of the penalty.

Let $P$ be a planar polygon, and let $f_P(n)$ denote the number of words of length $n$ generated by coding billiard orbits according to the domains of regularity they visit. Gutkin…