33 problems

Let $\mathcal F$ be a holomorphic foliation of $\mathbb{C}\mathbb{P}^2$, and let $L_1$ and $L_2$ be non-invariant algebraic lines. For a holonomy germ … write $S$ for the countable…

Let $A$ be a finite subset of $\mathbb{C}$, and let $f\in\mathcal{A}(\overline{\mathbb{C}}\setminus A)$ be a germ analytically continuable along every path in…

Let $k,q\in\mathbb{N}$, $v\in\mathbb{C}$, and $\omega,\lambda\in\mathbb{R}_{>0}^2$. Let $H^{q,k}(z,v,\omega,\lambda)$ be the integral defined by … where $J$ and $\mathfrak{g}$ are…

Isolated-barrier conjecture. In the case of nonlinear systems, infinitely many equally spaced isolated barriers should occur.

Levenberg–Poletsky–Merker conjecture. The closure of the analytic extension of $A$ coincides with $A^*_D$.

Soient $P,Q\in\mathbb C[x,y]$ et considérons l'équation différentielle … Supposons que ni $P$ ni $Q$ ne soient identiquement nuls et que $Q$ ne contienne aucun facteur vertical, de…

Artin's conjecture. The completed function $\Lambda(s,\psi)$ is entire. Equivalently, $L(s,\psi)$ can be analytically continued to $\mathbb{C}$ apart possibly from a pole at $s=1$…

Even-meson mass-vanishing conjecture. Under analytic continuation $\alpha\to\alpha^*_k$, the spectrum contains a pair of mesons whose masses vanish according to

Infrared CFT conjecture. Following Fateev, each of these points corresponds to a critical point in the infrared, whose dynamics are governed by a non-trivial conformal field theory…

Entire-periodization conjecture. The function $G(\sigma)$ is well defined and is an entire function of $\sigma$. This analytic property would justify the periodization used to rewr…

Let $r_n$ and $s_n$ denote the roots occurring in the AGM recursion, let $a_\infty$ and $u_\infty$ be its limiting quantities, and let $\sin_p^{-1}$ denote the principal value of t…

Consider a potential for which the roots of $x\widetilde V(x)=0$ are positive, distinct, and ordered as … Let the TBA equations be obtained by assuming that the logarithms of the r…

Let $\mathit{\Gamma}$ be a core graph, let $V_\mathit{\Gamma}$ be its vertex set, and for each 2-partition $p_2$ of $V_\mathit{\Gamma}$ let $D_{p_2}$ be the domain defined by the v…

Meromorphic-continuation conjecture. These twisted Weyl-group multiple Dirichlet series admit meromorphic continuation to the interior of $X_0^*$ and are holomorphic there except f…

For each topological type $w$ of spectral network, let ${\mathcal V}_w$ be the region where the WKB network has type $w$, let ${\mathcal X}_w$ be the associated coordinates, and le…

Let $S$ be non-decreasing on $[0,infty)$, and suppose that its Laplace-Stieltjes transform satisfies … Assume this function extends analytically to the half-plane…

Let ${}_{p}\Psi_q$ denote the Fox–Wright function, and consider the series in equations,, and, each expanded about the relevant point in the variable $1-z$. Radius-of-convergence c…

Exponential decay conjecture. The eigenvalues $\lambda_n$ and coefficients $\pi_n$ have these exponential decay asymptotics, and the asymptotic upper bound for the eigenvalues capt…

Asymptotic balance conjecture. As $\epsilon\to0$,

Let $A$, $B$, $C$, and $D$ be the operator entries in the block-operator setting, with $D$ self-adjoint and with spectrum $\sigma_D=\sigma_D^{\rm ac}=\overline{\Delta}$, where…

Wedge-of-the-edge conjecture. The function $f$ analytically continues to $(-1,1)^n$.

Let … be Pearson's $n$-step ramble integral, initially defined for $\operatorname{Re}s>0$ and analytically continued to complex $s$. For $n\in\mathbb Z_{>1}$, the only possible sin…

Logarithmic stability conjecture. For small $\varepsilon$ and large $N$, the error satisfies

Let $W^{\vee}_0=W^{\vee}$ be the Gepner point and $W^{\vee}_{\infty}=\prod_i x_i$ the large complex structure point in the deformation space of $W^{\vee}$. Let the corresponding Sa…

Consider the sine-Gordon model with $0<\beta^2\leq 1$, and let $f_a(\theta_1,\ldots,\theta_N)_{1\ldots1}$ denote relative form factors whose particles are all first breathers. For…