17 problems

Let $\mu$ be a CNA+ measure, and condition it on the total rank $\sum_eX_e$. Rank-cover conjecture. For each $k$, the conditional distribution given total rank $k+1$ stochastically…

Let $\mu$ be a measure on a Boolean lattice. It is ULC+ when every measure obtained from it by imposing an external field and projecting is ULC. ULC implication conjecture. If…

For a measure on a Boolean lattice, its rank sequence is the sequence of probabilities of the possible values of the total number of occupied coordinates. It is ULC (ultra-log-conc…

Let $\mu$ and $\nu$ be compactly supported probability measures, let $c\in[0,\infty]$, and let $\mu\oplus_c\nu$ denote their $c$-convolution. Let $f\in C^{4}(\mathbb{R})$ satisfy…

Limiting-measure and operator conjecture. The family $\hat\nu_\kappa^\sigma$ converges weakly as $\sigma\to0$ to a measure $\hat\nu_\kappa$ on \hat\mathcal A. Moreover, the opera…

Let $S_n$ be the symmetric group and let $\mathbb F_r$ be the free group on $r$ generators. For a word with constants $w\in S_n\ast\mathbb F_r$, let $\mu_w^{S_n}$ be the distributi…

Let $\mathcal{P}$ and $\mathcal{Q}$ be families of probability measures on $\mathfrak{X}$. Suppose there exist $P_0\in\mathcal{P}$ and $Q_0\in\mathcal{Q}$ such that $P\ll P_0$ for…

Weak convergence conjecture. On ${\mathcal P}_{\Gamma,q,\ell}$, the measures $nu_{q,\ell,n}$ converge weakly, as $n\to\infty$, to the measure $mu_{\Gamma,q}$ from the existence and…

Positivity conjecture. For every such $\a$ and $\b$, there exists a probability measure on $N$-tuples $\c=(c_1\le\dots\le c_N)$ satisfying the identity for all…

Let $M=(X,K,\phi)$ be a simplicial metric thickening, with metric thickening $\mathcal{M}\subseteq\mathcal{P}(X)$. Let $\hat{M}$ be the closure of $\mathcal{M}$ in $\mathcal{P}(X)$…

Limit-measure conjecture. The measures $mu_g$ converge to a measure $mu$ on $\mathcal{E}$ such that the expected number of surjections from a $mu$-random element to every element o…

Let $\nu_0$ be the probability measure induced by Werner's loop measure on the triangular factorization parameter $a$ of a welding homeomorphism, and let $\beta_0<\frac{5\pi^2}{4}$…

Let $\mathfrak{P}$ be the set of measures considered in the paper that arise directly or indirectly, after extension, from a Markov stopping time, and let Theorem be the cited cons…

Let $\mathfrak{P}$ denote the set of measures considered in the paper that arise directly or indirectly, after extension, from a Markov stopping time. Closure conjecture. The set…

Let $K\subset\mathbb{R}^{d}$ be a compact set, and let $\mathcal{M}(K)$ be the space of probability measures on $K$ with its weak topology. For a measure $\mu\in\mathcal{M}(K)$, wr…

Let $Q(z)$ and $P(z)$ be polynomials with $\deg Q(z)=l\geq 2$ and $\deg P(z)\leq l-1$, and let $\mathcal V_n$ be the finite measure on the space $Pol_{l-2}$ of monic polynomials of…

Let $P=(Z,<)$ be a causal set, let $\Omega$ be its space of natural extensions, and let $\mu$ be an order-invariant measure on the associated measurable space. An element…