15 problems

Let $X$ be a compact Riemann surface, let $P\subset X$ be a finite set, and let $(V,\nabla_V)$ be a semisimple holomorphic flat bundle on $X\setminus P$. Let $(H,D_V,h)$ be the tam…

Parabolic stability necessity conjecture. If $\widehat M$ admits a scalar-flat Kähler metric, then $E$ must be parabolically stable. This is a necessary-condition conjecture for th…

Let $N$ and $s$ be integers with $0<s<N$, and let … For $\alpha\in W(N,s)^{\circ}$, let $M(\underline{\mathbf{1}})^\alpha$ be the moduli scheme of quasiparabolic bundles with multi…

Kostov's conjecture. There is an irreducible solution to

Let $M$ be a blown-up ruled surface of genus at least $2$, and let $[\omega]$ be an admissible Kähler class, meaning that $c_1\cup[\omega]=0$. Associate to $[\omega]$ its correspon…

Let $X$ be a smooth proper curve over $\mathbb{F}_p$, let $G\to X$ satisfy the stated hypotheses, fix a parabolic subgroup $P$, and let $\mathcal{E}$ be an augmented commutative…

Periodic de Rham conjecture.

Weak periodic de Rham conjecture. The natural inclusion functor

Extension conjecture. The canonical parabolic extension of an irreducible periodic de Rham bundle over $U$ is again periodic.

Trace formula conjecture. One has

General geometric conjecture. There exists an action of $\mathfrak{H}$ on $H_c^i(\mathcal{X}_{\bf S}^{\mathcal{E}},\mathbb{C})$ such that

Geometric conjecture. (i) The quotient exists as an algebraic variety and $X_{\bf A}^{\mathcal{E}}\to\mathcal{X}_{\bf A}^{\mathcal{E}}$ is a principal…

Degree-independence conjecture. For every integer $d$, the sum

Deligne's conjecture. There exists a polynomial $A^{{\bf b};\mathbf{m}}_{\boldsymbol{\mu}}(t)\in\mathcal{C}h(S_{\boldsymbol{\mu}})[t]$, with coefficients in the positive cone…

Polynomiality conjecture. There exists a polynomial $A_{\boldsymbol{\mu};\mathbf{d}}^{\mathbf{m}}(t)\in\mathbb{Z}[t]$ such that for every finite field $\mathbb{F}_q$ and every divi…