28 problems

Let $m$ be the lattice-width parameter, let $q$ be the Potts-model state parameter, and let $\lambda_{\star,=}(m)$ and $\lambda_{\star,\neq}(m)$ denote the dominant transfer-matrix…

Let $f(w)$ be the spanning-forest free energy per site on a regular lattice of dimension $d\geq2$ and coordination number $r$, and let $f_0$ denote the entropy per site for spannin…

Let $P(t)=\sum_{j=0}^d\pi_jt^j$ satisfy $\pi_0=0$ and $\pi_1=\pi_2$, and set $p_0(x)=x$. Consider the formal expansion with recursively defined polynomial coefficients $p_k$, and l…

Let $P(t)=\sum_{j=0}^d\pi_jt^j$ satisfy $\pi_0=0$ and $\pi_1=\pi_2$, and let $p_0(x)=x$. Assume that the formal expansion has polynomial coefficients $p_k$ as in the polynomial sol…

Let $P(t)=\sum_{j=0}^d\pi_jt^j$ be a polynomial satisfying … and set $p_0(x)=x$. Let $p_k$ denote polynomials whose coefficients are determined recursively by the formal expansion…

Let $M$ be a compact Riemann manifold with negative curvature. Let $\Gamma$ be a finitely generated torsion-free nilpotent group, let $\Phi:\pi_1(M)\to\Gamma$ be surjective, and le…

Fix $c\in(0,1)$. Let $\mathcal L_D^c$ be a Brownian loop soup of central charge $c$, let $\mathcal C$ be one of its clusters, and let $:\mathtt h_c\Theta_{\mathcal L_D^c}^{n-1}:$ d…

Let $M$ be a closed and oriented $3$-manifold. For each positive integer $k$, let $Z_k(M)\in\mathbb{C}$ denote the $\operatorname{SU}(2)$ WRT invariant for $M$ at level $k$. Let…

Let $d_2(B)$ be the function associated with the Heun equation, the confluent Heun equation, and the reduced confluent Heun equation, and let $D_k$ and $D_k^{[j],(k+j+1)}$ denote t…

Let series 33 in Table have $t=-1$, and let $\mathcal{R}_n$ and $\mathcal{F}_n$ be the quantities associated with this series. Let $c_j$ be the coefficients defined by Theorem. Env…

Let $L_n^{\varoast}$ denote the relevant longest monotone subsequence length for $\varoast\in\{\boxslash,\boxbslash,\boxdot\}$, and let $\gamma(\varoast)$ be the corresponding scal…

Let $L_n^{\varoast}$ denote the relevant longest monotone subsequence length for $\varoast\in\{\boxslash,\boxbslash,\boxdot\}$, let $\gamma(\varoast)$ and $\delta(\varoast)$ be the…

Let $k\geqslant4$, and let $a_{k,j}$ be the coefficients in the asymptotic expansion, with parameters $\alpha$, $\varkappa$, $\varepsilon$, $b$, and $a_{1,\pm1}$ as defined in the…

Let $a_{k,j}$ be the coefficients in the asymptotic expansion indexed by $k\in\mathbb{N}$ and $j\in\mathbb{Z}$ with $|j|\leqslant k$. Parity and extremal-coefficient conjecture. If…

Topological expansion conjecture. There exists a positive constant $\varepsilon$ such that, for each nonnegative integer $k$,

Let $\beta$ be the expansion parameter, let $a$ be the regularization parameter, and let $\psi_{a,\beta}(x)$ denote the function given by Theorem. At fixed $a$ and $x$, consider an…

Let $M$ be a symmetric homogeneous mean that has an asymptotic power series expansion, and suppose that it fulfills the requirements of the open question from Farhi. Let $L_c$ deno…

Let $a_k$ and $b_k$ be the quantities whose asymptotic expansions are under consideration, and let the $N$th error term mean the error obtained by stopping the corresponding expans…

Let $j_{\nu,\kappa}$ denote the $\kappa$th positive zero of the Bessel function $J_\nu$, and let McMahon's expansion be the asymptotic expansion for these zeros. For…

Let $r$ and $j$ be parameters, let $z_i$ be positive integers, and let $c_i$ be coefficients. Define … with $a_0=1$. The Awesome conjecture. The coefficients of $\ln(F)$ satisfy ……

Derrida–Hilhorst conjecture. Suppose first that there exists $\alpha\in(0,+\infty)$ such that $\mathrm{E}[Z^\alpha]=1$. If $\alpha\notin\\{1,2,\ldots\\}$, then, as $\epsilon\to0$,

Granath's conjecture. It holds that

Long-distance expansion conjecture. The sine-Gordon/Painlevé $\mathrm{III}_3$ tau function is given by the convergent series

Berndt–Kim conjecture. For sufficiently large $n$, the coefficients $a_n$ have the same sign.

Let $(v_n)_{n\geq 0}$ be the sequence of shifts appearing in the asymptotic expansion … The sequence begins with $v_1=\frac{23}{90}$, $v_2=\frac{1792627}{7289730}$, and…