12 problems

Real-monodromy oper conjecture. The Langlands parameters should be in correspondence with the set of $^LG_\mathbb C$-opers on $X$ having real monodromy, after restricting to single…

Gaiotto's classification conjecture. Real opers are classified topologically by the real locus of their developing map.

Let $X$ be a smooth projective irreducible curve over a local field $F$, let $G$ be a connected reductive algebraic group over $F$, and let $S$ be a finite set of $F$-points in $X$…

Let $F=\mathcal{C}$, let $X$ be a smooth projective curve over $F$, let $G$ be a reductive group, and consider the commuting Hecke operators acting on the Hilbert space of half-den…

Let $\operatorname{Op}^\mathrm{alg}_{{}^LG}(X)_{\mathbb R}$ be the set of ${}^LG$-opers on $X$ with real monodromy, and let $\operatorname{Spec}(\mathcal A_G)$ be the joint spectru…

Let $G$ be a reductive group, let $X$ be a curve over a local field $F$, and let $\lambda\ne0$. Write $V_G(\lambda)$ for the dense domain on which the Hecke operators are initially…

Let $X$ be a curve over $\mathcal{C}$, let $G$ be a reductive group, and let $H_\lambda(x)$ be the Hecke operators acting on the Hilbert space $\mathcal H_G$ of half-densities on…

Basic spectral decomposition conjecture. The Hilbert space has an orthogonal decomposition

Teschner's real-monodromy conjecture. The corresponding eigenvalues are parametrized by $PGL_2$-opers with real monodromy.

Third analytic Langlands conjecture. The set $\Sigma$ is contained in the set of ${}^LG$-opers on $X(\mathbb C)$ defined over $\mathbb R$.

Second analytic Langlands conjecture. The spectrum $\operatorname{Spec}_{\mathcal A}(\mathcal H)$ is discrete; equivalently, $\mathcal H$ is the completed direct sum of finite-dime…

First analytic Langlands conjecture. The pair $(\mathcal A_{\mathbb R},S(\mathcal A))$ is a strongly commuting family of unbounded essentially self-adjoint operators on…