51 problems

Positivity conjecture for motivic Chern classes. For every $u\leq w\in W$,

Quotient conjecture. This composition is an isomorphism.

Let $G$ be the compact Lie group and $EG$ the universal principal $G$-bundle, with $Ad(EG)$ its adjoint bundle and $T_v$ its vertical tangent bundle. Let $LBG^{-TBG}$ denote the as…

Let $T$ be a maximal torus in $G$, let $\mu$ be an orbit of representations of an arbitrary quiver of type $A$, and let $\overline{\mu}$ denote its closure. A K-theoretic lace diag…

Let $G$ be a connected reductive algebraic group over ${\mathbb{C}}$ with simply-connected derived group, let $\tilde{W}_a$ be its extended affine Weyl group, and let…

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

Let $G$ be a rank-one compact Lie group with maximal torus $T$, let $\mathfrak{M}$ be the moduli stack of $G$-bundles, and let $\mathfrak{M}_T$ be the corresponding moduli stack of…

Let $G$ be a compact Lie group, let $\Sigma$ be a closed surface of genus $g$, and let $\mathfrak{M}$ be the moduli stack of $G$-bundles on $\Sigma$. Let $\Pi:G^{2g}\to G$ be the p…

Let $G$ be a compact Lie group, let $\mathfrak{M}$ be the moduli stack of $G$-bundles over a genus-$g$ surface, let $\tau_0$ and $\sigma$ be the twistings appearing in the index co…

Let $G$ be a complex reductive group with nilpotent cone $ilcone$, let $KG( ilcone)$ be the Grothendieck group of $G$-equivariant coherent sheaves on $ilcone$, and let…

Relative fixed-point formula conjecture. The diagram

Vistoli's conjecture. If

Let $\mathcal{R}$ be the coefficient ring, let $P_{\mathrm{loc}}=P\otimes_{\mathcal{R}}\operatorname{Frac}\mathcal{R}$ for every $\mathcal{R}$-module $P$, and let … be the unique i…

Let $\mathcal{L}(n,r)$ be the specified closed projective subvariety of the Gieseker space, let $K_{\mathcal{T}}(\mathcal{L}(n,r))$ denote its equivariant $K$-theory, and let the c…

Localization conjecture. $D$ satisfies the localization condition with respect to every separable compact $G$-$\mathrm{C}^*$-algebra $B$.

Let $\mathfrak{g}$ be a reductive Lie algebra with Weyl group $W$, let $\mathfrak{C}_W$ be the set of two-sided cells in $W$, and let $\mathbf{a}:\mathfrak{C}_W\to\mathbb{N}$ be Lu…

Hausel's K-theoretic conjecture. The ring of invariant functions is, as an algebra over $\mathbb C[\mathrm G]^{\mathrm G}\cong K^0_{\mathrm G}(\operatorname{pt})\otimes\mathbb C$,…

Let $G$ be a compact group. The equivariant Kasparov theory $KK^G$ is said to be very special when it has the special properties defined in the surrounding discussion. Compact-grou…

Flag-manifold relations conjecture. For $\ell$ in the geometric window, the $\prod_i S_{k_i}$-invariant combinations of the Coulomb-branch equations generate the ideal of relations…

Grassmannian relations conjecture. Symmetric combinations of the equations

Balmer's comparison-map conjecture. For every finite group $G$, the comparison map is a homeomorphism

Countable stratification conjecture. The category $\Cell{G}$ is countably stratified for every finite group $G$.

Let $X=Fl(v_1,\dots,v_k;N)$ be the partial flag variety of flags $V_1\subset\dots\subset V_k\subset\mathbb{C}^N$ with $\dim(V_i)=v_i$. Let $\mathcal{S}_i$ be the tautological subbu…

Let $\mathrm G$ be a principally paired group acting regularly on a smooth projective variety $X$. Define the fixed point scheme $\operatorname{Fix}_{\mathrm G}(X)$ by the conditio…

Equivariant classification conjecture. There is an order-preserving $\mathbb Z[x,x^{-1}]$-module isomorphism