87 problems

The universal-enveloping-algebra expression conjecture. For every $n$,

Let $n\geq 4$, and let $({\mathcal{M}}_t)_{t\in[0,\infty)}$ be the birth Mallows process. If $T_k$ are its jump times, its jumping process is…

Let $\mathcal C_k$ denote the limit shape of the random-core growth process for fixed $k$. Let $D_k$ be the piecewise-linear curve with vertices … where $\gamma$ is the scaling con…

Symmetry of complements. For all $\lambda\in\mathcal R_k$, $\pi(\lambda)=\pi(\lambda^{\mathrm{comp}})$. The conjecture is supported in the paper by computations for $k\leq 6$.

Negative partial-sum conjecture. If there are pairwise different indices $i_1, \dots, i_{n-1}$ such that

Polynomial mixing-time conjecture. For any graph $G$, the Glauber dynamics on the $k$-colourings of $G$ for $k\geq\Delta+2$ has polynomial mixing time.

Let $n$ particles evolve according to the Attracting Random Walks (ARW) model on an arbitrary graph, with interaction parameter $\beta<0$. The mixing time is measured with respect…

Let $(X_n)_{n\in\mathbb{Z}}$ be a stationary reversible Markov chain with transition operator $Q$, invariant distribution $\pi$, and let…

Let $M$ be an indecomposable, doubly stochastic matrix indexed by an alphabet of size $k$, satisfying … where $k+1\equiv1$. A word of length $N$ is generated by the Markov chain wi…

Contingency-table norm conjecture. For arbitrary table dimensions $m$ and $n$, table sum $\Sigma$, and row and column sums $r_k$ and $c_p$, this multi-decomposition is non-degenera…

Exact spectral-gap conjecture. The inequality relating the $m$-urn and two-urn gaps is an equality:

Let $\mu$ be a Hurwitz-stable distribution supported on one parity class. Consider the two-step parity flip--repair graph and its walk on each connected component: the walk flips a…

Let $n\geq 3$ and let $\vec{p}$ be a regular parameter vector. A label is neutral if $p_{c,i}=p_{i,c}=1/2$ for every $i\neq c$. Let … be the number of neutral labels, and let…

Fill's characterization conjecture. The following are equivalent:

Connectivity conjecture. For every $n$, the set of reduced bumpless pipe dreams of size $n$ is connected under the $2\times2$ flips that preserve reducedness.

Skewed sampling conjecture. There should be a polynomial-time algorithm that approximately samples uniformly from the set of $\vec{n}$-colorings of $G$ whenever

Fix $k$, and consider the Burnside chain on $(C_k^n,S_n)$ for varying $n$. Let $\lambda$ be any nonzero eigenvalue of this chain. The binomial multiplicity conjecture. There exist…

Consider the spiked Wigner inference problem with signal-to-noise parameter $\beta_{\bM}$ and threshold $\beta_{\bM}$ as used in the paper. Let $\beta_{\bM}$ denote the scaled post…

In the spiked Wigner inference problem, let annealed Glauber dynamics refer to the Markov chain studied in the paper, and let weak recovery mean achieving nonzero correlation with…

Fix $k\ge 2$, and for each $n$ let $K_n$ be the Burnside process on $(C_k^n,S_n)$. Consider starting states whose limiting empirical distribution on $C_k$ assigns a positive propor…

Fix $k$, and let $\lambda$ be any nonzero eigenvalue of the Burnside chain on $(C_k^n,S_n)$ for any $n$. Multiplicity binomiality conjecture. The eigenvalue $\lambda$ occurs with m…

Let $e,f,h$ be the basis elements of $\mathfrak{sl}_2$, and define the $2\times 2$ matrices … For nonnegative integers $x,y,z$, let $f(x,y,z)$ be the sum over all orders of taking…

Fix an arbitrary order on the positions of the corners of a Rubik's cube. Let $X_t$ denote the configuration of the cube at time $t$, and call two corners unlinked when they are un…

Let $M$ be the position matrix of the one-shelf shuffling machine. Its eigenvalues describe the associated Markov chain on card positions; the previously established eigenvalues in…

The quadratic eigenvector conjecture. $M$ has an eigenvalue-eigenvector pair $(\frac{1}{4},\vec{v})$ with multiplicity at least $1$. Numerical computation suggests this eigenvector…