109 problems

Baum–Connes conjecture for discrete groups. The analytic assembly map

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

Let $\Gamma$ be a discrete group. Write $\underbar{E}\Gamma$ for its classifying space for proper actions, let $KO^\Gamma_*$ denote equivariant $KO$-homology, and let…

Let $T$ be a torus, let ${\mathcal F}$ be its flag manifold, and let $W$ be the Weyl group. For the opposite Schubert varieties $X^u$ and $X^v$, write their equivariant structure-s…

Vistoli's conjecture. If $\alpha\in\ker\tau_X$, then $\xi\alpha=0$ for some class $\xi$ of a perfect complex with everywhere non-zero rank.

Let $\Delta_{\lambda,\mu,\nu}$ be the triangular puzzle region whose left, right, and bottom boundary labels are respectively $\lambda$, $\mu$, and $\nu$. A KV-puzzle is a filling…

Quotient conjecture. This composition is an isomorphism.

Product decomposition conjecture. The rationalized ring $K_0(X,G)\otimes {\mathbb Q}$ should split as a product

Let ${\mathcal O}$ be a nilpotent orbit with associated parameter $\lambda_{\mathcal O}$, and let $R_{+0}$ and $R_{+1}$ be the positive roots of gradation $0$ and $1$, respectively…

Let ${\mathcal O}$ be a nilpotent orbit, let $a({\mathcal O})=\frac12\operatorname{codim}_{\mathcal N}{\mathcal O}$, and let $j:{\mathcal O}\to {\mathcal N}$ be the inclusion of th…

Let ${\mathcal O}_\lambda$ be the nilpotent orbit whose closure is the support of $C(\lambda)$, and let $j_{{\mathcal O}_\lambda}:{\mathcal O}_\lambda\hookrightarrow {\mathcal N}$…

Let $G$ be a simple simply connected algebraic group, let ${\mathcal N}$ be its nilpotent cone, and let $K_{\mathcal G}({\mathcal N})$ have canonical basis elements $C(\lambda)$ in…

Let $G$ be a simple simply connected algebraic group, let ${\mathcal N}$ be its nilpotent cone, and let ${\mathcal G}=G\times {\mathbb C}^*$ act on ${\mathcal N}$ as above. Let…

The paper considers the operators appearing in Theorem 2.1, acting at the level of equivariant $K$-theory. Rigidity conjecture. The operators considered in Theorem 2.1 are rigid on…

Köck's conjecture. There is a push-forward of completions

Let $G$ be an algebraic group, let $\Psi$ be a semisimple conjugacy class in $G$, choose $h\in\Psi$, and let $Z={\mathcal Z}_G(h)$. For a $G$-space $X$, write…

Let $X$ be a $G$-space, let $\Psi$ be a semisimple conjugacy class in $G$, choose $h\in\Psi$, and let $Z={\mathcal Z}_G(h)$. Theorem (c) states that the map…

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 $X$ be the affine flag manifold, let $X_w$ be the Schubert variety indexed by $w\in W$, and let $\mu$ be a weight. Write $\mathscr{O}_X(\mu)$ and $\mathscr{O}_{X_w}(\mu)$ for t…

The paper studies equivariant and nonabelian symplectic quotients in integral $K$-theory, where the results avoid torsion difficulties that obstruct analogous statements in integra…

Let $G_0$ and $G_1$ be countable abelian groups, let $\alpha_i$ be automorphisms of $G_i$ of prime orders $p_i$, and let $H_i\subseteq G_i$ be $\alpha_i$-invariant subgroups satisf…

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…

Let $G$) be a connected reductive algebraic group over ${\mathbb{C}}$ with simply-connected derived group, let $\tilde{W}_a=WP$ 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