300 problems

Let $f:({\mathbb C}^{m+1},0)\to({\mathbb C},0)$ be an isolated quasihomogeneous hypersurface singularity, let $(H_{\mathbb Z},M)$ be its associated integral lattice with monodromy,…

Nagai's conjecture. The monodromy operators satisfy

Monodromy Conjecture. If $s_0$ is a pole of $Z_{f,W}^{\mathrm{mot}}(s)$, then $(s-s_0)\mid b_f(s)$, and $e^{s_0\cdot 2\pi \sqrt{-1}}$ is an eigenvalue of a local monodromy at some…

Let $F=(f_1,ots,f_r)$ be a tuple of pairwise irreducible polynomials, let $f=f_1\cdots f_r$, and let $\textbf{1}=(1,\dots,1)$. For a point $\mathfrak{x}$, write…

Toledano Laredo's monodromy conjecture. The monodromy of the trigonometric Casimir connection $\widehat{\nabla}_C$ should be described by the action of the affine braid group $B_G$…

Let $K$ be the local field, let $X$ be a proper smooth variety over $K$, and let $H^j(X^{\mathrm{ca}},\overline{\mathbb{Q}}_l)$ carry its monodromy filtration $M_\bullet$ and weigh…

Let $\mathfrak g$ be a Lie algebra with Cartan subalgebra $\mathfrak h$, and consider the De Concini–Procesi connection on the set of regular elements of $\mathfrak h$. Its horizon…

Let $D\subset\mathbb{CP}^2$ be an algebraic singular plane curve with cusps and nodes, and let $\theta:\pi_1(\mathbb{CP}^2-D)\to S_N$ be a symmetric-group-valued monodromy morphism…

Monodromy generation conjecture. The monodromy representation is generated by the operator $\mathbf{B}_{\mathbf{m}}$.

Gauge-invariance conjecture. The contour integral is the generating function for the eigenvalues $\mathcal{I}^{(2p-1)}_{\ell,\{t_i\}}$, and these eigenvalues are unchanged under th…

Let $W$ be the Weyl group associated with a singularity of type $D_n$ or $E_n$. For some proper smooth surface $X/K$, consider an integral model of $X$ whose special fiber has a si…

Let $f_0$ be an isolated hypersurface singularity, and let $M^{mar}_\mu$ denote the moduli space of marked singularities in its $\mu$-homotopy class. Connectedness conjecture. The…

Let $A$ be a discrete valuation ring with finite residue field $k$, let $X$ be a regular scheme projective and flat over $A$ with strict semistable reduction, and put…

Denef and Loeser's motivic monodromy conjecture. There exists a finite subset $\mathcal{S}\subset\mathbb{Z}\times\mathbb{Z}_{>0}$ such that

Let $F$ be a face of the secondary polytope, with one vertex corresponding to the smooth Calabi--Yau phase $W$. Let ${\mathcal E}^\bullet(F)$ be the kernel associated with the tori…

Consider a classical confluenting family of hypergeometric equations, with a nonperturbed equation having an irregular singularity. Let the perturbed equation have appropriately no…

An irregular singularity of a linear differential equation has Stokes operators, and a deformation of the equation has monodromy whose limiting behaviour can be considered as the d…

Topological cyclotomic Hecke algebra isomorphism conjecture. The topological cyclotomic Hecke algebra is conjectured to be isomorphic to the cyclotomic Hecke algebra in all cases.

Let $\mathcal X\rightarrow S$ be a family of Calabi–Yau manifolds, let $X=\mathcal X_t\xrightarrow{f}B$ be a torus fibration with a section $\delta_0$, and let $X^\sharp$ be the co…

Let $S\subset {\mathbb P}^n$ be a primitive K3 surface, and let $d>0$. Consider the family of rational curves in the linear system $|{\mathcal O}_S(d)|$. Transitive monodromy conje…

Let $f$ be a polynomial and let $X_s=f^{-1}(0)$ be its complex zero locus. The naïve motivic zeta function $Z_{mot}(f,s)$ is the motivic analogue of the $p$-adic and topological ze…

Let $\pi:\mathcal X\to\Delta$ be a degeneration of an irreducible symplectic $2n$-fold. For each $m$, let $T_m$ be the associated monodromy operator on…

Let $\pi:\mathcal X\to\Delta$ be a degeneration of an irreducible symplectic $2n$-fold. For each $m$, let $T_m$ be the associated monodromy operator on…

Let $f$ be a singular function, with braid monodromy group acting by the Artin representation on its algebraic monodromy homomorphism. Stabilizer conjecture. The braid monodromy gr…

Let $X$ be a variety with totally degenerate reduction, and let $N_{1,L}$ and $N'_{1,L}$ be the enriched monodromy operators defined in Section 4.3. Write…