38 problems

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

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 $S$ 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…

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-…

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

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…

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 $G$ be a reductive group, $G^{L}$ its Langlands dual, and $M$ a closed oriented $3$-manifold. Kapustin–Witten Langlands duality conjecture. There is an isomorphism … This is mo…

Generic-oper contractibility conjecture. The space of generic oper structures on a given irreducible local system is homologically contractible.

Let $\mathcal C$ be a braided fusion category over $\Bbb C$. For objects $X_1,\ldots,X_n,Y\in\mathcal C$, the space $\operatorname{Hom}(Y,X_1\otimes\cdots\otimes X_n)$ carries an a…

Let $S$ be a finite subset of $\boldsymbol{P}^1$ and let $P$ be a quasiunipotent Riemann symbol for $S$, meaning the list of eigenvalues of monodromies around points of $S$. A conn…

Let $X$ be a smooth projective complex variety, and let $\mathcal{L}$ be an irreducible rigid local system on $X$, meaning that it cannot be deformed continuously or algebraically…

Let $\mathbb L$ be a geometric local system on a smooth variety $X$, with underlying vector bundle with connection $(E,\nabla)$. For all sufficiently large primes $p$, require that…

Let $S$ be the base scheme, let $X$ be the relevant smooth variety over $S$ with boundary divisors $D_i$, and let $T_i$ prescribe the monodromy at infinity. Suppose there is one, o…

Let $\mathcal{A}$ be a complex hyperplane arrangement in $\mathbb{C}^n$, and let $M(\mathcal{A})$ be its complement. For a nonzero $\omega\in H^1(M(\mathcal{A}),\mathbb{Z}_2)$, wri…

Cohen–Dimca–Orlik type conjecture. The cohomology groups satisfy

Budur–Wang's special varieties package conjecture. The following conditions hold in $M_B(X,r)$: (A1) absolute $\overline{\mathbb{Q}}$-constructible subsets are generated from absol…

Integer-finite local-system conjecture. Then $L$ is an integer finite local system, and there is an isomorphism

Special-locus density conjecture. The set of complex points of $Z$ corresponding to representations of geometric origin is dense in $Z$. In particular, the special points are dense…

Density conjecture. The set of $G$-representations of geometric origin is Zariski dense in

Let $k$ be a field, let $X$ be a smooth projective curve over $k$, and let $G$ be a reductive group over $k(X)$ with Langlands dual group $\LG$. Let $S\subset X$ be a finite set. F…