31 problems

Let $(X,g)$ be a closed, $2n$-dimensional Riemannian manifold with negative sectional curvature $\mathrm{sec}_{g}$. Let $h^k_{(2)}(X)$ denote the $k$-th $L^2$-Betti number of $X$.…

Let $\mathcal M_k^0$ be the reduced moduli space of $\operatorname{SU}(2)$ monopoles of magnetic charge $k$ on $\mathbb R^3$, and let $\widetilde{\mathcal M}_k^0$ be its universal…

Let $G$ be a countable group with non-vanishing first $\ell^2$-cohomology. Equivalently, suppose that there is a function $f:G\to\mathbb{C}$ with $f(0)=0$, which is not in…

Davis–Okun conjecture. If the $i$th $L^2$-Betti number of $G$ is nonzero, then

Let $V$ be a complex projective variety, let $M=V\setminus\operatorname{sing} V$ be its regular part, and equip $M$ with the metric induced by a smooth Kähler metric on projective…

Cheeger–MacPherson $L^2$-duality conjecture. The group $H^i_{(2)}(U)$ is always dual to $IH_i(X)$, and the pairing is given by integration.

Let $S$ be a noncompact smooth curve with smooth compactification $\overline S$, let $j:S\hookrightarrow\overline S$ be the inclusion, and let $L_{\rho}$ be the local system associ…

Let $M$ be a manifold with boundary fibration … and suppose that the metric on $M$ is quasi-isometric near the boundary to … where $R\in[1,\infty)$, $-1\leq c<0$, and $k$ is a symm…

Bueler's conjecture. The weighted $L^2$ cohomology of $M$ with respect to $d\mu$ is isomorphic to the de Rham cohomology of $M$.

Let $X$ be a Shimura variety with special cycles $\mathcal Z(m)$ whose generating series is modular in cohomology, and let $X^*$ be its Satake–Baily–Borel compactification. Looijen…

Let $X$ be a Shimura variety, let $X^*$ be its Baily–Borel compactification, and let ${\mathbb E}$ be a polarized variation of Hodge structure naturally defined over $X$. Write…

Lück's approximation conjecture. Under the stated conditions, the displayed equality holds. This is a special case of the conjecture that is stated in the source as already proved…

Let $\mathfrak M_{\zeta}$ be a Nakajima quiver variety equipped with its Nakajima metric, and suppose that the parameter $\zeta$ is generic. Let…

Eyssidieux's conjecture. These filtrations endow

Let $X$ be a complex projective manifold, let $\mathbb{M}$ be a Mixed Hodge Module on $X$, and let $H^k_2(\widetilde{X},\mathbb{M})$ denote the associated $L_2$ cohomology, equippe…

Let $Z$ be a singular compact complex space, let $\pi:\widetilde{Z}\to Z$ be its universal covering space, and write $\mathcal{IC}^{\bullet}_Z$ for the intersection cohomology shea…

Let $\operatorname{Hilb}_0^n(\mathbb C^2)$ be the centered Hilbert scheme of $n$ points on $\mathbb C^2$, equipped with the Nakajima metric. Write…

Let $X$ be a compact Kähler space, let $X^o\subset X_{\rm reg}$ be a dense Zariski open subset, let $(\mathcal{V},\nabla,\mathcal{F}^\bullet,h)$ be an $\mathbb{R}$-polarized variat…

Let $M$ be a $\rm II_1$ factor and let $Q_1,Q_2\leq M$ be von Neumann subalgebras such that $Q_i\leq M$ is approximately $L^2$-rigid for $i=1,2$. Suppose that $Q_1\cap Q_2$ is diff…

Let $G$ be a countable group. A Bernoulli action of $G$ is an action on a product probability space $(X,\mu)=(\{0,1\},\mu_0)^G$ obtained from a product measure, not necessarily inv…

Let $X$ be a Hermitian locally symmetric space with Baily-Borel compactification $X^*$, let $\mathbf E$ be a local system arising from a representation of the ambient reductive gro…

Let $\mathcal{M}$ be the moduli space of Higgs bundles, and let $H_c^*(\mathcal{M})$ and $H^*(\mathcal{M})$ denote its compactly supported and ordinary cohomology, respectively. Su…

Let $\Gamma$ be a discrete group with well-defined $\ell^2$-Betti numbers $b_i^{(2)}(\Gamma)$, and let $\operatorname{actdim}(\Gamma)$ denote its action dimension. If…

Let $M$ be a closed aspherical Riemannian manifold and let $\Gamma=\pi_1(M)$. Let $\mathcal{N}\Gamma$ denote the group von Neumann algebra of $\Gamma$, and let $l^2\Gamma$ denote t…

Cohomology coincidence conjecture. The $L^2$ cohomology associated to a self-dual mezzoperversity and the cohomology of a self-dual sheaf coincide.