18 problems

Topological multivariable strong monodromy conjecture. The polar locus of $Z_{F}^{\operatorname{top}}(S)$ is contained in $V(B_{F})$.

Let $f:({\mathbb C}^{n+1},0)\to({\mathbb C},0)$ be a germ of an analytic function, and let $Z_{top,0}(f,s)\in{\mathbb Q}(s)$ be its local topological zeta function. A pole $s_0$ of…

Let $\mathsf{M}$ be a loopless matroid with topological zeta function $Z^{\operatorname{top}}_\mathsf{M}(s)$. A Mengesha–Miranda–Sun conjecture. The topological zeta function has a…

Let $f\in \mathbb C[x_{1},\dots,x_{n}]\setminus \mathbb C$ be a non-constant polynomial, let $d\in\mathbb Z_{\geq 1}$, and let $Z^{\mathrm{top},(d)}_{(f)}(s)$ denote the twisted to…

Topological invariance conjecture. The function $Z_{f,\bm 0}^{\mathrm{top}}(s)$ depends only on $(\bm w,d)$, and is a topological invariant of the family $\mathbf{QH}_n$.

Monodromy conjecture. If $s_0\in\mathbb C$ is a pole of $Z_{f,\operatorname{top}}(s)$, then $e^{2\pi i s_0}$ is an eigenvalue of $T_f$ and $s_0$ is a zero of $b_f(s)$.

The multivariate monodromy conjecture. If $F$ is such a tuple, then

Let $f\in\mathbb{C}[x_1,\dots,x_n]\setminus\mathbb{C}$ with $f(0)=0$, let $Z_{\operatorname{top},0}(f;s)$ be the local topological zeta function at the origin, and let $b_{f,0}(s)$…

Let $f$ be a polynomial with coefficients in a field of characteristic $0$, and let $Z_\mathrm{top}(s)$ be its local topological zeta function. If $\alpha$ is a pole of…

Topological monodromy conjecture. If $\theta$ is a pole of $Z_f^{\mathrm{top}}(s)$, then $\exp(2\pi i\theta)$ is an eigenvalue of the monodromy $M_x^{(q)}$ for some $q\in\mathbb N$…

Strong Monodromy Conjecture. The poles of $Z_I(s)$ are roots of the Bernstein–Sato polynomial $b_I(s)$; in the stronger form,

Let $\mathbb{Q}[\varepsilon_n]=\mathbb{Q}[x]/(x^n)$, and let $\mathbf{H}[\varepsilon_n]$ denote the group scheme obtained from the Heisenberg Lie algebra tensored with…

Let $f$ be a polynomial and let $Z_f(s)$ be the topological zeta function attached to the divisor $\operatorname{Div}(f)$. If…

Let $F=(f_1, 3ellots,f_r)$ with $0\ne f_j\in\mathbb{C}[x_1, 3ellots,x_n]$, let $\mu:Y\to X$ be a log resolution of $\prod_jf_j$, and write $a_{i,j}$ for the order of vanishing of…

Let $f:(\mathbb C^{n},0)\to(\mathbb C,0)$ be a germ of a non-zero holomorphic function, and let $Z^{}_{\operatorname{top,0}}(f,s)$ and $b_{f,0}(s)$ denote its local topological zet…

Igusa–Denef–Loeser conjecture. The poles of $Z^{}_{\operatorname{top,0}}(f,s)$ are roots of the local Bernstein-Sato polynomial $b_{f,0}(s)$. This relates poles of topological zeta…

Let $f\in\mathbf{C}[x_1,\ldots,x_n]$ be a non-constant polynomial with $f(0)=0$, let $\mu:Y\to\mathbf{C}^n$ be a log resolution of $f^{-1}(0)$, and write … for its local topologica…

Let $\mathcal{I}$ be an ideal defining a subscheme, let $\pi$ be an embedded resolution of $\mathcal{I}$, and let $Z_{\mathrm{top},\mathcal{I}}^{(d)}$ denote the associated topolog…