68 problems

Root-of-unity conjecture. The eigenvalues of $M_i$ must be roots of unity.

Let $G$ be a reductive group and let a local system on a smooth curve be irreducible and rigid, with quasi-unipotent local monodromies and finite-order abelianization. Simpson's co…

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…

Let $X$ be a smooth connected projective algebraic curve over $\mathbb{C}$, let $G$ be a reductive group over $\mathbb{C}$, and let…

Let $X/K$ be a smooth projective geometrically connected curve over a number field, let $D\subset X$ be a reduced divisor, and set $U:=X\setminus D$. Let $L$ be a rank $r\geq 1$ ir…

Let $G$ be an algebraic group, and let $\mathrm{Char}_G$ be the character variety parametrizing $G$-local systems. A local system is motivic when it is of geometric origin. Esnault…

Let $G$ be a simple simply connected algebraic group, let ${\mathcal N}$ be its nilpotent cone, and let ${\mathcal G}=G\times {\mathbb C}^*$ act on ${\mathcal N}$ as above. Let…

Generic local-system cohomology conjecture. For every $p$ and almost all $a\in A_1^*$ among those with

Rigidity conjecture. Any quasi-unipotent $G$-rigid local system on $X$ is motivic: there exist a smooth morphism $\pi:Y\to X$ and an integer $i$ such that the local system is a sub…

Hartl's local-system maximality conjecture. The space $\breve{\cal{F}}^0_b$ is the largest open $\breve E$-analytic subspace of $\breve{\cal{F}}^{wa}_b$ carrying a local system…

Let $G$ be a reductive group with Langlands dual group $\cG$, let $E_\cT$ be a $\cT$-local system on $X$, and let $\DefbE$ be the DG formal scheme parametrizing its universal Borel…

Let $G$ be a reductive group with Langlands dual group $\cG$, let $E_\cT$ be a $\cT$-local system on $X$, and let $\DefbE$ be the DG formal scheme parametrizing deformations of…

Let $G$ be a reductive group, $X$ a smooth complete curve, $\cG$ the Langlands dual group, and $E_\cG$ a $\cG$-local system on $X$. A Hecke eigensheaf is an object…

Let $S$ be the surface under consideration, let ${\cal X}_{PGL_3,S}$ be its moduli space of framed $PGL_3$-local systems, and let $n$ be a positive integer. A good positive Laurent…

Achar–Sommers' local-system conjecture. The Dynkin diagram of $\mathcal O$ is assigned by Lusztig's bijection to a pair $({\mathcal O}',\rho)$ such that $\rho$ occurs in

Let $X$ be a complex variety with a codimension-two stratum … whose transverse singularity is a normal surface singularity with local discriminant package $(E,q)$. Then the higher-…

Local systems on the moduli space $\mathcal{M}$ of Riemann surfaces are determined by monodromy representations … where $\pi_1^{\mathrm{orb}}(\mathcal{M})$ is the mapping class gro…

Let $M$ be a manifold. Let $\mathrm{LocSysCat}(M)$ denote the $(\infty,2)$-category of local systems of categories on $M$, and let $^{2}\mathrm{Fuk}(\mathrm{T}^{*}[1]M)$ denote the…

Let $\overline{k}$ be a separable algebraic closure of a finitely generated field $k$, let $f:X\to S$ be a smooth proper morphism of smooth $\overline{k}$-schemes, and let…

Let $\overline{k}$ be a separable algebraic closure of a finitely generated field $k$, let $f:X\to S$ be a smooth proper morphism of smooth $\overline{k}$-schemes, and let…

Let $C$ be the curve in question, let $\Theta$ be the sheaf whose pullback $i^!\Theta$ occurs in the twisted global geometric Langlands correspondence, and let $\mathcal E$ be a sy…

Structural decomposition conjecture. The stack $\operatorname{LS}_G^{\mathrm{restr}}(D^{\circ})$ is a disjoint union of formal completions of a collection of pairwise non-intersect…

Gaitsgory's factorization conjecture. There exists a fully faithful functor

Let $X$ be a smooth irreducible complex projective variety, let $(E_i)_{i\in S}$ be a finite set of mutually distinct prime divisors on $X$, and let $\bm a=(a_i)_{i\in S}$ with…

Let $X$ be a smooth projective complex variety and let $\omega^{1/q}$ be a multivalued rational $n$-form on $X$ with no logarithmic poles and whose divisor has normal-crossings sup…