74 problems

Berry–Keating conjecture. When expanded to its optimal order, the Riemann–Siegel formula should approximate $Z(t)$ with an error of order

Yang's conjecture. The coefficients satisfy

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 $\mathcal{K}$ be a hyperbolic knot. Let $N,k\rightarrow\infty$ with $u=\frac{N}{k}$ fixed, and consider the colored Jones polynomial…

Granath's conjecture. It holds that

Let $W(x)$ denote the function studied in the paper, and let $(2a)$ and $(4c)$ be the two displayed expansions for $W(x)$ introduced earlier. Convergence conjecture. The series…

Let $\overline M$ be a smooth compact manifold with boundary, let $M$ be its interior, and let $g$ be a conformally compact Einstein metric on $M$. A smooth conformal infinity is t…

Let $X$ be a closed oriented $3$--manifold, let $G$ be a simply connected compact simple Lie group with complexified Lie algebra $\mathfrak g$, and let $\tau_{r}^{\mathfrak g}(X)$…

Consider the superunitary Calogero–Moser–Sutherland model with coupling parameters $(\beta_1,\beta_2)$, whose wavefunction admits an asymptotic expansion analogous to the Hankel ex…

The paper considers Yang–Mills gauge potentials whose asymptotic expansion is … with coefficients depending only on $\tau$ and $a$, and imposes the conditions obtained from the Yan…

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 closed oriented 3-manifold, let $\mathcal{M}(M,\mathrm{SU}(2))$ be the moduli space of flat $\mathrm{SU}(2)$ connections over $M$, and let $\mathrm{CS}[\gamma]$ denote…

Let $\ell$ be a natural number, and consider the small-$k^2$ expansions of the eigenvalue branches associated with the operator $\Delta_K-k^2 {\mathscr A}$. Their radius of converg…

Ren and Li's conjecture. For all $x>0$ and all integers $m\geq1$,

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…

Conjectured coefficients.

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…

Positivity conjecture. Each polynomial $c_n(x)$ has only positive coefficients. According to the source, this would follow from the corresponding positivity conjecture for the poly…

Let $t_n(x)$ be the integer-coefficient polynomials defined by … and, for $n\geq 1$, … Here $(x)_m$ denotes the rising factorial. Positivity conjecture. Each polynomial $t_n(x)$ ha…

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…