58 problems

Meromorphic invariant potential conjecture. Every constant mean curvature surface in $\mathbb{R}^3$ associated with $M$ can be obtained from an invariant meromorphic potential.

Let $G$ be the group considered in the paper, let $BG$ be its classifying space, and let $\mathcal{L}BG$ denote its free loop space. Write $G(\mathbb{F}_q)$ for the corresponding f…

Gaitsgory–Kazhdan irreducibility conjecture. The object $\Pi\in \operatorname{Rep}(\BG)$ is irreducible.

The Verlinde algebra isomorphism conjecture. For every homogeneous degree $z^n$ with $n\neq 0$, the map

Let $Sp(n)$ be the compact symplectic group, let $U(i)$ be the unitary group, and let $K.\Omega Sp(n)$ denote the K-homology ring of its based loop space. For $1\leq i\leq n$, the…

Garland–Murray's conjecture. Periodic instantons can be constructed using holomorphic spheres in a flag manifold associated to the loop group.

$B_1$ distribution conjecture. The distribution of $B_1$ is

Let $\nu$ be a $L_{pol}SU(2)$-invariant probability measure on $\bold L(SL(2,\mathbb C)/SU(1,1))$, and let $g_0$ be the zero-mode component with Cartan and matrix coordinates…

Let $\mu$ be the conjectured unique invariant measure on the relevant loop-group completion, and let $\theta_{-}$ denote its negative-mode coordinate. For every matrix coefficient…

Let $G$ be the complexification of a simply connected compact Lie group, and let $L_{hyp}G$ be the hyperfunction completion of its loop group. Let $\mathcal D$ be the group of anal…

Let $X$ be a simply connected compact symmetric space with $X\simeq U/K$, where $U$ is a simply connected compact group acting isometrically and transitively on $X$, and let $G$ be…

Let $K$ be a simply connected compact Lie group, let $G$ be its complexification, and let $L_{pol}K$ act by left and right multiplication on the formal completion $L_{formal}G$ of…

Zero-mode distribution conjecture. The pushforward measure is

Freed–Hopkins–Teleman isomorphism conjecture. The map constructed by Segal is conjecturally an isomorphism.

Let $\mathfrak{W}_{\mathbb Q}$ be the Wiener algebra of absolutely summable rational-frequency Fourier series, and let $\mathfrak{W}_{\mathbb Q}^{\pm}$ denote its positive- and neg…

Assume that $G$ is of type A. Let $\widetilde{\mathrm{KP}}_0$ be the full subcategory of the Koszul perverse coherent heart generated under extensions by the objects in…

Let $G$ be a reductive group over $\mathbb{C}$, let $\check{G}$ be its Langlands dual group, and let $G_{\mathcal{O}}$ and $\check{G}_{\mathcal{O}}$ denote their positive loop grou…

With $X$, $G$, $\ul{x}$, and $\ul{\operatorname{Shv}}_{\operatorname{HL}}(\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}})$ as above, regard this object as an object of…

Let $X$ be a smooth complete curve over $k$, let $\ul{x}$ be a finite set of $k$-points, and let $\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}}$ be the moduli stack of $G$-b…

Let $G$ be a reductive group and let $\fL(G)\operatorname{Cat}_{\operatorname{restr}}$ be the restricted 2-category of categories acted on by its loop group. Let…

Let $G$ be a reductive group with Langlands dual $cG$, let $\fL(G)$ be its loop group, and let $\fL(G)\operatorname{Cat}_{\operatorname{restr}}$ be the restricted 2-category of cat…

Let $G$ be a reductive group, let $\xi=(M,\lambda,L,\mathcal{O},\mathcal{L})$ be a polar-cuspidal datum for $G$, and let $\mathbf{Q}\subset LM$ be a parahoric subgroup containing…

Representability conjecture. The fpqc sheaf $\prescript{}{1}{\mathcal{C}_1^{\leq\mu}}$ is represented by a normal $k$-scheme. This predicts that the indicated truncated double quot…

Centralizer transitivity conjecture. The group $(LG)_{\mathfrak g}$ acts transitively on $\operatorname{Gr}_{\mathfrak g,\mathcal O}$. The paper notes that this fiber is nonempty a…

Let $g\in VMO(S^1,SU(2))$. Let $A(g)$ and $A_1(g)$ be the operators associated with $g$, and let a triangular factorization mean the factorization specified in (II.3). A root subgr…