25 problems

Let $G$ be a connected reductive group, let $G_\eta$ range over its pure inner forms, and let $\mathbb{L}^u_{\underline q}$ be the graded unipotent Langlands parameter space. Write…

Let $\mathbb{X}_{\mathbf{p},\underline{\lambda}}$ be a weighted projective line, let $\mathfrak{Lg}$ be its corresponding loop Kac–Moody algebra, and use the identification…

Let $X$ be a separated scheme over $\mathbb{C}$ of finite type and let $A$ be an abelian group. The semi-topological Quillen-Lichtenbaum conjecture for coherent sheaves. The canoni…

Faithfulness conjecture. For any $\alpha\in({\mathbb Z}^2)^+$, the polynomial representation $P^{\frac12}_\alpha$ of $A^{\frac12}_\alpha$ is faithful.

Let $X$ be a two-sided Hecke pattern of rank $r$. Write $H_*(X)$ for its Borel–Moore homology, and let $W_{\star,\star}(S)$ denote the algebra with the appropriate choices of indic…

For a regular semisimple $\gamma$, define the graded module … It is a module over the graded algebra $\oplus_{d=0}^{cinfty}{}_0\mathcal A_d$ and therefore defines a quasi-coherent sh…

The construction associates to each $gammabin g(cK)$ a {cC}^-equivariant quasi-coherent sheaf $cFgamma$ on $cmathfrak C{ccheck G}$. Coherence conjecture. For any regular semi…

Let $f:Y\to X$ be a proper morphism between smooth projective algebraic varieties. Assume that the degenerate locus of $f$ is contained in a normal crossing divisor $D\subset X$. L…

Let $K$ be a nonarchimedean local field, let $d$ be a positive integer, and let $]_{ll]}$ be the coefficient ring. Let $L]_d(K)$-representations mean representations on $]_{ll]}$-m…

Let $G$ be the reductive group under consideration, let $\widetilde{G}$ denote its Grothendieck resolution, and let ${\mathcal N}$ be the nilpotent cone in its Lie algebra. Write…

Formal K-flatness conjecture. Formal K-flatness is equivalent to K-flatness.

Let $X$ be a ribbon, let $\delta=-\deg({\mathscr N})$, and let $\bar{g}$ be the genus of $X_{\mathrm{red}}$. The large-nilradical component conjecture. If … then the only irreducib…

Let $X$ be a ribbon, let $r$ be a positive integer, and let $\delta=-\deg({\mathscr N})$, where ${\mathscr N}$ is the conormal sheaf of $X_{\mathrm{red}}$ in $X$. Fix the generaliz…

Let $X$ be a ribbon and let ${\mathscr F}$ be a pure sheaf on $X$ that is not a generalized vector bundle. A sheaf generizes to another sheaf if it occurs as a special fiber of a f…

Suppose we are in the reductive case. Let $boldsymbol{ u}in-boldsymbol{X}^+$, and let $({mathscr{O}}{boldsymbol{ u}},boldsymbol{\rho}{boldsymbol{ u}})inXi$ be the pair corresponding…

Finite-type extension conjecture. The category $\mathcal{P}_{coh}^{G(\mathcal{O}) \rtimes \mathbb{G}_m}(\mathrm{Gr}_{G_{Ad}})$ is a monoidal categorification of the quantum cluster…

Let … and let $\vec{\operatorname{gr}}(\mathcal{T})$ be the associated graded object of the Hodge-module kernel $\mathcal{T}$. Dense-open line-bundle conjecture. There exists a den…

Let $\mathcal M^{\rm mix}$ be the graded subcategory from the Hodge-theoretic Koszul duality conjecture. Let $\kappa$ be the functor from $K$-equivariant coherent sheaves on…

Let $G_{\mathbb R}$ be a real group with Cartan decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$, let $\mathcal M_{\rm Ho}$ be the full subcategory of $K$-equivariant Hodge…

Let $(B,H,C)$ be a homology triplet, and let $\mathcal{E}^{\bullet}$ be a complex of coherent sheaves corresponding to it as in the realization conjecture. Hilbert-polynomial uniqu…

Let $(B,H,C)$ be a homology triplet of type $n$, with degrees of $B$ given by … and strand starts of $H$ and $C$ given respectively by … A complex of coherent sheaves…

Let $X$ be a complex manifold, let $Y\subset X$ be a subvariety, and write … for the open embedding. Let $\mathcal{A}_X$ denote the relevant sheaf of microdifferential operators, a…

Assume the Strominger–Yau–Zaslow fibration conjecture holds, let $(X,W)$ be the relevant mirror-symmetry data, and let $(L,b)$ be a Lagrangian brane. For each point $p\in\hat X$, i…

Let $\mathcal{S}_{n-k,k}$ be the Slodowy slice, let $Y$ be the corresponding Springer fiber, and let $A$ range over all irreducible components of $Y$. Write…

Let $\mathcal{C}$ be the heart of the $t$-structure on coherent sheaves on the smooth compactification $Z_n$, and let $A$ be the corresponding highest-weight algebra. Coherent-shea…