12 problems

Positivity conjecture. The Green's function $H_{m,M}$ is positive if and only if $M\in(-m,\lambda_1)$.

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 $d_1$ 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…

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…

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…

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…

Let $\mathbf G$ be a group of type $A$, let $\mathbf L\leq\mathbf G$ be an $F$-stable Levi subgroup, and let $v\in\mathbf L_{\operatorname{uni}}^F$. Let $u$ be the induced class of…

Quantitative Green's-function convergence conjecture. There exists a positive constant $C<\infty$, depending on $d$, $\mathfrak{p}$, $\lambda$ and $s$, such that for every $t>0$ an…

Let $M$ be an arbitrary complete connected Riemannian manifold, let $B(o,r)$ denote the geodesic ball centered at $o$, let $\mu$ be the Riemannian measure, let $G(x,o)$ be the Gree…