17 problems

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…

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…

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 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 $X\subset \mathbb{P}^N$ be a projective variety, let $X_{\rm reg}$ be its regular locus, and let $ds^2_{\rm FS}$ be the Fubini–Study metric on $X_{\rm reg}$. Write…

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 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 $\Gamma$ be an $n$-dimensional Poincaré duality group, and let $b_i^{(2)}(\Gamma)$ denote its $i$th $\ell^2$-Betti number. Singer conjecture. The $\ell^2$-Betti numbers vanish…

MacPherson's conjecture.

Cheeger–Goresky–MacPherson conjecture. There is a natural isomorphism

Vafa–Witten conjecture.

Let $\bar{\M}^\mu_\B$ be the character variety quotient associated with a genus-$g$ Riemann surface and parabolic type $\mu$, and let $\chi_{L^2}(\bar{\M}^\mu_\B)$ denote its topol…

Let $M^{k,c_1}_\phi$ be the smooth completion of the moduli space of ${\rm U}(n)$ Yang--Mills instantons of first Chern class $c_1$, energy $k$, and framing $\phi$ on a four-dimens…

Let $M_k^0$ be the hyperkähler moduli space of certain ${\rm SU}(2)$ magnetic monopoles on $\mathbb R^3$, and let $\widetilde{M}_k^0$ be its universal cover. Write…