36 problems

Let $\mathcal{M}_{\B}^d$ be the $\operatorname{GL}_n$ character variety with central monodromy $\zeta_n^d I_n$, and let $\hat{\mathcal{M}}_{\B}^e$ be the corresponding…

Let $n$ be a positive integer, let $d,e$ be integers satisfying $(d,n)=(e,n)=1$, and let $\mathcal{M}_{\Dol}^d$ and $\hat{\mathcal{M}}_{\Dol}^e$ denote the corresponding Dolbeault…

The geometric Langlands correspondence. There exists a canonical equivalence of categories

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, let $X$ be an irreducible smooth projective variety over $\mathbb{K}$, and let $G$ be a reductive group ove…

Let $(A,\phi,\psi)\in\mathcal{M}_\alpha^G$ be a solution and let $b=\mathrm{Hit}_\alpha^G(A,\phi,\psi)$. Denote by $\Sigma_b$ the associated spectral curve and by $L$ the correspon…

Let $C$ be a smooth projective curve, let $M_{r,d}$ be the moduli space of semistable Higgs bundles on $C$, and let $\mathbb G_m$ act by $\lambda\cdot(E,\phi)=(E,\lambda\phi)$. For…

Let $V_{g,n}$ be the vertex operator algebra associated with the genus-$g$ theory with $n$ regular punctures, and let $\{L_a\}$ be its simple modules. Let ${\cal M}_{g,n}$ be the m…

Let $V_{g,0}$ be the vertex operator algebra associated with the unpunctured genus-$g$ theory, with simple modules $\{L_a\}$ and conformal dimensions $h(L_a)$. Let ${\cal M}_{g,0}$…

Let $X=G/H$ be an affine homogeneous spherical variety with abelian regular centralizers and no type $N$ roots, and let $S_X$ be a polarizable symplectic dual representation of…

Let $\mathcal M$ be a derived Hitchin stack, let $\operatorname{Sht}_{\mathcal M}$ be its associated shtuka stack, and let $c_1$ be the correspondence map whose derived dimension i…

Multiplicative motivic perverse filtration conjecture. The motivic perverse filtration is multiplicative with respect to cup-product:

Let $M(n,0)$ be the degree-zero moduli space in the Hitchin fibration, with Hitchin map $\chi(n,0)$ to its basis. The complex $R\chi(n,0)_*\operatorname{IC}(M(n,0),\mathbb{Q})$ is…

de Cataldo–Hausel–Migliorini P=W conjecture. Under this identification, the perverse filtration associated with $\pi$ satisfies

Let $c\in C$ and let $\mathrm G$ be a reductive group. For $\mu\in X_+(\mathrm G^\vee)$, let $\rho_\mu$ be the corresponding irreducible representation of the Langlands dual group,…

Let $C$ be a smooth projective curve, let ${\mathcal M}$ be the smooth quasi-projective $\SL_n$ Hitchin moduli space, and let ${\mathcal M}^\vee={\mathcal M}/\Gamma$ be the quasi-p…

Let $\mathcal A$ be the Hitchin base. A basic WKB cameral network associated to $a\in\mathcal A$ consists of a suitable homotopy-equivalent open subset of the curve together with f…

Odd orthogonal limiting metric conjecture. The solutions to the decoupled Hitchin equation for $\SO(2n+1,\C)$ constructed in Theorem are limiting metrics.

Limiting metric conjecture. The solution $h_{dc}(E,\Phi,\omega)$ to the decoupled Hitchin equation is a limiting metric: $h_t$ converges to $h_\infty$ in $C^\infty$ on every compac…

Let $G$) be a reductive group, and let $M_\mathrm{H}$ and $M_\mathrm{B}$ be the corresponding Hitchin and Betti moduli spaces for a curve $C$. Let $h:M_\mathrm{H}\to\mathbb{A}^d$ b…

Let $({\overline{\partial}}_E,\varphi)$ be a Higgs bundle in the regular locus $\mathcal M'$, and let $g_{L^2}$ and $g_{\mathrm{sf}}$ denote Hitchin's $L^2$-metric and the semiflat…

Let ${\mathcal M}_{\rm par}$ be the moduli space of singular parabolic Higgs bundles with $n$ simple poles in their connection, and let ${\mathcal M}_{\rm wild}$ be the moduli spac…

Let a compact singular elliptically fibered Calabi–Yau $3$-fold have singular fibers associated to ${\mathcal G}$-symmetry in codimension $1$ over the base, admit a crepant resolut…

Let $\bm{\mu}$ be the multipartition encoding the parabolic data of $M_B(n)$, let $k$ be as defined in the paper, and let $\mathbb{H}_{\bm{mu}}$ be the function defined in Section…

Let $G$ be a semisimple complex Lie group, let $X$ be a Riemann surface, let $\mathcal C$ be the associated $3$-dimensional Calabi--Yau category, and consider the component of its…

Let $G$ be a semisimple complex Lie group, let $X$ be a Riemann surface, let … be the $3$-dimensional Calabi--Yau category associated to the singularities of the harmonic map to th…