20 problems

Let $G$ be a group, let $E_{\mathcal{VCY}}(G)$ be the classifying $G$-space for the family of virtually cyclic subgroups, and let $\mathcal{A}$ be an additive $G$-category with inv…

Let $\mathcal{P}$ be the Procesi bundle on $\operatorname{Hilb}^n(\mathbb{C}^2)$, let $\mathcal{P}^*$ denote its dual, and let $\mathcal{O}(d)$ be the indicated determinant-line-bu…

Let $G$ be a group and let $\mathcal C$ be a class of subgroups of $G$. There is a $G$-CW complex $E_{\mathcal C}(G)$ whose fixed-point set is contractible for subgroups in…

Let ${\mathcal H}^?_*$ be an equivariant homology theory with values in $R$-modules, let $G$ be a group, and let ${\mathcal C}$ be a family of subgroups of $G$. Let…

Let $G$ be a group, let $d53c$ be a family of subgroups of $G$, let $E$ be a functor from small $d53d$-linear categories to spectra that sends category equivalences to weak equival…

Let $G$ be a connected reductive group, let $G({\mathbf O})$ act on the affine Grassmannian $\operatorname{Gr}_G$, and let $\mathbb G_m$ act by loop rotation. Write $\check G$ for…

Let $\Gamma$ be a countable group acting properly and isometrically on the Euclidean space $\mathbb{E}_{\Gamma}$, and let $\mu_{\mathrm{IP}}$ denote the assembly map for the equiva…

Let $Y$ be a rational homology sphere, let $E$ be the bundle used to define the equivariant instanton complex, let $R$ be the coefficient ring, and let $\sigma$ be a choice of sign…

Stable-envelope existence and uniqueness conjecture. There exist unique classes $(S_p)_{p\in\mathfrak M_1^T}$ in $H_{\dim\mathfrak Z}^T(\mathfrak M_1)$ such that, for every…

Meta-Isomorphism Conjecture. The assembly map is bijective for all $n\in\mathbb{Z}$. This framework includes the Farrell--Jones and Baum--Connes conjectures after suitable choices…

Let $Y$ be a rational homology sphere equipped with a spin structure $\mathfrak s$. Let $\mathit{HFI}^+(Y,\mathfrak s)$ denote the corresponding variant of involutive Heegaard Floe…

Let ${\mathbf S}\colon\operatorname{SPACES}\to\operatorname{SPECTRA}$ respect weak equivalences and disjoint unions. Let $G$ be a group and ${\mathcal F}$ a family of subgroups of…

Let $\Gamma$ be a group, let $(G,\psi)$ be a group over $\Gamma$ with homomorphism $\psi\colon G\to\Gamma$, let ${\mathcal H}^?_*$ be an equivariant homology theory over $\Gamma$,…

Let ${\mathbf S}\colon\operatorname{SPACES}\to\operatorname{SPECTRA}$ be a covariant functor respecting weak equivalences and disjoint unions. Let ${\mathcal C}$ be a class of grou…

Let ${\mathbf S}\colon\operatorname{SPACES}\to\operatorname{SPECTRA}$ be a covariant functor respecting weak equivalences and disjoint unions. Let $G$ be a group, ${\mathcal F}$ a…

Let ${\mathbf S} \colon \operatorname{SPACES}\to \operatorname{SPECTRA}$ be a covariant functor respecting weak equivalences and disjoint unions. For a group $G$, let…

Let $\cF$ be a family of groups, let $G$ be a countable discrete group, and let $W$ be a free $G$-CW-complex. Let $E_\cF G$ denote the classifying space of $G$ for the family $\cF$…

Let $G$ be a group, let $\mathcal{F}$ be a family of subgroups of $G$, let $E$ be a functor from small $\mathbb{Z}$-linear categories to spectra that sends category equivalences to…

Let $G$ be a group, let $mathcal F$ be a nonempty family of subgroups of $G$ closed under conjugation and under taking subgroups, let $E$ be a functor from small $mathbb Z$-linear…

Meta-isomorphism conjecture. The assembly map $A_{\mathcal{F}}$ is an isomorphism for every $n\in\mathbb{Z}$.