39 problems

Let $R\subset\mathbb{C}$ be a finitely-generated $\mathbb{Z}$-algebra, and let $S$ be a smooth $R$-scheme. Let $(\mathscr{E},\nabla)$ be a flat vector bundle on $S/R$. Vanishing…

Nekrasov–Rosly–Shatashvili conjecture. The superpotential $\mathcal{W}^{\mathrm{eff}}$ may be obtained as a generating function of opers in terms of these Darboux coordinates.

Unitarity conjecture for the Casimir connection. The connection $\nabla_{\kappa}$ ought to be unitary for any $\mathfrak g$. This is suggested by its identification, in type $A$, w…

Let $B_{\mathfrak{g}}$ be the generalised braid group, let $V[\negthinspace[\hbar]\negthinspace]$ be the formal deformation of a $\mathfrak{g}$-module $V$, and let $\mathcal V$ be…

The paper constructs invariants of tangles with flat connections over their complements using quantized universal enveloping algebras at roots of unity. Equivalence conjecture. The…

Consider generic holomorphic $\operatorname{SL}(N,\mathbb{C})$-connections on $C_{0,2}$ with irregular singularities of the mildest possible type at $0$ and $\infty$. Let…

Let $M$ be a closed oriented 3-manifold, and consider analytically continued Chern–Simons theory with complex level $k$. The relevant flat connections include flat $\mathrm{SU}(2)$…

Let $Y$ be the complement of a normal crossing divisor in a smooth proper Deligne–Mumford stack. A flat bundle on $Y$ is quasi-geometric if it extends as an algebraic regular-singu…

Bundles with flat connections associated to braided fusion and modular fusion categories are defined over $\overline{\mathbb{Q}}$. Etingof–Varchenko's conjecture. For any braided f…

Degree conjecture. In order to properly count asymptotic ends of $\bar{\mathcal{M}}_{\rm flat}^{\rm irr}$ corresponding to the singular points that obstruct the almost-surjective m…

Let $M$ be an oriented three-dimensional manifold and let $G=\operatorname{SL}(2,\mathbb{C})$. The moduli space of flat $G$-connections on $M$ may have infinite asymptotic ends, ar…

Let $R\subset\mathbb{C}$ be a finitely generated $\mathbb{Z}$-algebra, let $f:X\to S$ be a smooth proper morphism of smooth $R$-schemes, fix $s\in S(R)$, and let $\widehat{S}$ and…

Let $X/\mathbb{C}$ be a smooth projective curve, let $D\subset X$ be a reduced effective divisor, and let $(\mathscr{E},\nabla:\mathscr{E}\to\mathscr{E}\otimes\Omega_X^1(\log D))$…

Let $R\subset\mathbb{C}$ be a finitely generated $\mathbb{Z}$-algebra, let $X$ be a smooth $R$-scheme, let $(\mathscr{E},\nabla)$ be a flat vector bundle on $X/R$, let…

Finite-set correspondence conjecture. The connected components of the anyonic $SL(N,\mathbb C)$ flat-connection space are in one-to-one correspondence with the points of

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 $X$ be a smooth algebraic variety over $\overline{\mathbb{Q}}$ and let $\mathscr{F}$ be a flat bundle on $X$. A $G$-connection is a flat bundle satisfying the arithmetic condit…

Behnke's conjecture. Every reflexive module on a cusp singularity admits a flat connection.

Let $D\subset\mathbb{C}^{k}$ be a hypersurface cut out by a reduced holomorphic function $f$, and let $r$ be the restriction-and-holonomy map … Here $f^{-1}(S^{1})$ is the inverse…

Let $(\overline{\partial}_E,\varphi)$ be a stable nilpotent Higgs bundle, and let $\Phi$ be the Higgs field associated with the leading term of the gauge-equivalent family of flat…

Let $\mathcal{M}_\text{dR}$ be the moduli space of flat connections equipped with the partial oper stratification, whose leaves include the open-stratum leaves described by…

Let $\Sigma$ be a surface, and let $\mathcal{A}(\lambda)$ be a family of $\operatorname{PSL}_n(\mathbb{C})$-connections satisfying … with…

Higher non-abelian Hodge conjecture. There is a unique, up to unitary gauge, flat connection satisfying: locally $\Phi=\Phi_1\,dz+\Phi_2\,d\bar z$, with $\Phi_1$ principal nilpoten…

Let $U$ be a neighborhood of the zero-section in T^*\bm\hat{\mathcal{T}}^n, and let $[(\mu_k,t_k)]\in U$. A connection … is considered up to unitary gauge. Local existence and un…

Let $\mu$ and $t$ be the higher-complex-structure data, and consider a connection … Here $\Phi$ is induced by the higher complex structure, and $A$ is the connection term. Uniquene…