48 problems

Let $d_1$ and $d_2$ be discriminants such that one of them is fundamental when $k$ is even. For any CM points $z_1$ and $z_2$ of discriminants $d_1$ and $d_2$, let $H$ be the compo…

Higher-order mass contribution conjecture. The higher-order contributions in the mass to the perturbed Green's functions should be of higher order on the light cone.

Quasi-metric inequality conjecture. In this setting, the inequality holds, with the obvious modifications, for bounded Lipschitz domains $\Omega$ and operators $L$ with bounded mea…

Let $L$ be a differential operator with bounded measurable coefficients on a domain $\Omega$, and let $G$ be its Green function. Suppose the associated transformed kernel has the q…

Let $R_0$ be a root system of type $B_n$, let $\bbH$ be the associated graded Hecke algebra with long-root label $k_1\neq 0$ and short-root label $k_2=mk_1$, where…

Green-function normal-derivative conjecture. As $4y4$ approaches $4x4$ along the normal,

Strong Green function rigidity conjecture. If there exists such a point $p\in M$, then $M$ must be a round sphere.

Let $E_{\tau}$ be the rectangular torus associated with a fixed parameter $\tau$, and let $G_p(z)$ denote the Green function with parameter $p\in E_{\tau}$. Conjecture A. Fix any…

Let $G_p(z;\tau)$ be the Green function considered in the paper, with $p$ and $\tau$ as in the preceding results. Conjecture A. If $G_p(z;\tau)$ has $10$ critical points, then thes…

Let $K_\alpha$ be the kernel of the resolvent operator $(D^\alpha+2)^{-1}$ on $\mathbb R$, with $\alpha\in(2,4)$. Le–Pelinovsky conjecture. The kernel $K_\alpha$ is sign-changing f…

Let $T>1$, let $\hat{J}$ be the interval used in the definition of the Green's function, and let $\overline{H}_{m,M}(t,s)$ be that function for parameters $m$ and $M$. Let…

Let $S^\vee_a(x,y)$ be the causal Green's operator of the Klein–Gordon equation with mass parameter $a=m^2$, and let $\tilde{s}^\vee(x,y)$ be a perturbed Dirac Green's operator wit…

Let $G$ be a connected reductive group over a finite field, and let $G^F$ be its finite group of fixed points under a Frobenius map. Deligne–Lusztig define Green functions on the u…

Zero-background-modulus conjecture. The situation of a free-standing elastic sphere is simply found by setting the elastic modulus of the supporting background to zero.

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^3$, let $\lambda_1$ be the first eigenvalue of the Dirichlet Laplacian on $\Omega$, and for $\lambda\in(0,\lambda_1)$ let…

Green-function formula conjecture. In critical dimension, the Green function of $P_n$ should be

Green-function maximum-point conjecture. In more general domains, the point $x_M=x_M(q)$ may vary depending on $q$.

Let $\Gamma$ be a finitely generated nilpotent group, let $\mu$ be a symmetric admissible finitely supported probability measure, and let $S$ be a finite generating set. Writing…

Consider the fractional Anderson operator … on $\mathbb{Z}^d$, with $0<\alpha<1$, where $\lambda>0$ is the disorder parameter and the random potential has i.i.d. compactly supporte…

Drăgănescu et al.'s conjecture. The discrete Green's function satisfies

Let $\Gamma=SL_2(\mathbb Z)$, let $G_s(z_1,z_2)$ be the Green's function, and let … For a weakly holomorphic modular form $f=\sum_m c_f(m)q^m$ of weight $-2j$, define … For a discr…

Let $\Omega\subset\mathbb{R}^3$ be a smooth bounded domain, let $a$ be critical in $\Omega$, and let $G_a$ be the Dirichlet Green's function of $-\Delta+a$ normalized by … with reg…

Real-line Green's function conjecture. For every $c>0$ and every $\alpha\in(2,4]$, $G_{\mathbb{R}}$ is not strictly positive on $\mathbb{R}$ and is not monotonically decreasing on…

Periodic Green's function conjecture. For each $\alpha\in(2,4]$, there exists $c_0>0$ such that, for $c\in(0,c_0)$, $G_{\mathbb{T}}$ is even, strictly positive on $\mathbb{T}$, and…

A non-Hermitian system is purely dissipative when all eigenvalues of its Bloch Hamiltonian lie in the lower complex plane. Purely dissipative-system conjecture. Purely dissipative…