6 problems

Let $\Omega$ be a convex domain and let $Z_\Omega(s)$ be its tropical zeta function. The image of the map $\Omega\mapsto Z_\Omega$ consists of functions holomorphic on $\Re(s)>1$.…

Let $G_o(\lambda)$ be the Green function of the Laplacian on a symmetric space of noncompact type, and let $\widetilde{\mathcal Y}_{\pi/2}$ be the Riemann surface to which it conti…

Let $f$ be a polynomial, and let $\varphi$ be a test function whose support is sufficiently small. Gel'fand's conjecture. The integrals $Z_f(\varphi)(s)$ can be analytically contin…

Let $V_{n,m}$ be the Stiefel manifold, let $f$ be an integrable function on $V_{n,m}$, and let $\omega\in C^\infty(V_{n,m})$. Define … where $Q^\alpha$ is the analytically continue…

Let $f\in C^\infty(V_{n,m})$, and let $T^\alpha_{m,k}$ denote the cosine or sine transform under consideration. For $\omega\in C^\infty(V_{n,k})$, define the distribution-valued me…

Let $h(X_1,X_2)\in\mathbb{Z}[X_1,X_2]$ and define … A polynomial $g(U)$ is cyclotomic here if it divides $(1-U^{m})^{n}$ for some positive integers $m$ and $n$. Rudnick–du Sautoy c…