19 problems

Characteristic-cycle conjecture. The cycle $C_\lambda$ equals the characteristic cycle of the sheaf $\mathcal F_\lambda$. This identification would provide a geometric way to compu…

Let $E$ be a local field, and let $\rho:{\bf G}\rightarrow\operatorname{GL}({\bf V})$ be a nice representation of a simply connected semisimple group ${\bf G}$. Write…

Let $E=\mathbb{R}$, let $\rho:{\bf G}\rightarrow\operatorname{GL}({\bf V})$ be a nice representation, and assume that either ${\bf G}$ is simple or ${\bf H}$ is semisimple. Stable…

Aubert–Zelevinsky duality conjecture. If $X=X(q^{\frac{1}{2}{\it h}},\mathcal{S},\mathcal{L})$ is an irreducible unipotent representation in…

Let $G$ be a reductive group and let $\check G$ be its Langlands dual group. Denote by $\mathcal{H}_G^{dR}$ and $\mathcal{H}_{\check G}^{dR}$ the de Rham potent Hecke monads, and b…

Let $n$ be an even integer. An elementary wild character variety is a wild character variety arising from an elementary connection, considered up to isomorphism. Finiteness conject…

The Fourier-transform construction gives isomorphisms between wild character varieties, and for an irregular class $\mathit{\Theta}$ let $\widehat{\Theta}$ denote its formal Fourie…

Let $W$ be an irreducible spetsial complex reflection group, and write $\operatorname{Irr}(W)$ for its irreducible characters. Let $\operatorname{Deg}_\chi(q)$ and…

Let $G$ be a possibly disconnected reductive group, let $\mathcal N_G$ be its nilpotent cone, and let $\mathcal O$ be a nilpotent orbit. If…

Power-of-two coefficient conjecture. For any $x,y\in\bV$, either $c_{y,x}=0$ or

Let $C$ be a conical hypersurface and let $g(u_0,\ldots,u_n)$ be an admissible function on it. Assume that the rank $rk(C)$ is finite. Rational-form conjecture. Any admissible $g$…

Let $(G,V)$ be a visible stable polar representation for a connected reductive group $G$. Let $\pi\colon V\to V/\!/G$ be the quotient map, let $P$ be its nearby cycles sheaf, let…

Let $G$ be the reductive $p$-adic group and let $\mathsf{R}_{\mathsf{un},\mathsf{ell}}$ denote its elliptic unipotent representation space. Let…

Let $\widehat{\mathbb{C}}^{n+1}$ and $\mathbb{C}^{n+1}$ be dual affine spaces, and let…

Complexity conjecture. The number of steps required to compute $X_\alpha$ should not be much smaller than $\Omega(n\log n)$. This conjecture asserts that computing $X_\alpha$ canno…

Let $N=2n+1$, and let $E_{i,2j-1}^{2n+1}$ be the odd-indexed local systems in the paper's family. Write $\mathfrak{F}$ for the Fourier transform, $\operatorname{IC}$ for the inters…

Fourier decomposition conjecture. There exists such an $L$ inducing a canonical splitting

Let Fourier transform act on regular holonomic $\mathcal{D}$-modules in the setting of the Riemann sphere, with rigidity measured by Katz's cohomological numerical index. Katz's ri…

Extension conjecture. The isomorphism above should hold over the whole of $\operatorname{Bun}_{{\mathbb P}_n}$. This predicts that the explicit geometric realization of the minimal…