12 problems

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 $h$ be a complex polynomial and let $Z_{\mathrm{top},0}(h,s)$ be its local topological zeta function at the origin. If $s_0$ is a pole, write it in the source's notation as…

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 $f$ 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 $q\in\mathbb N$…

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

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 $f\in\mathbf{C}[x_1,\ldots,x_n]$ 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…