26 problems

Vanishing conjecture. We have

Burghelea–Haller conjecture. The Burghelea–Haller quadratic form satisfies

Let $C$ be a smooth closed curve with canonical constant-curvature metric $g_{0}^{C}$, and let $l$ be a holomorphic line bundle with induced canonical metric $g_{0}^{l}$. For a Her…

Let $C$ be a compact complex curve with Hermitian metric $g^{C}$, and let $l$ be a line bundle over $C$ with associated Hermitian metric $g^{l}$. Write $A(g^{C},g^{l})$ for the log…

Let $\mu:Y\to X$ be the birational modification in a semistable reduction, let $f:Y\to T$ be the induced family, let $F^{*}\xi$ be the pulled-back coefficient bundle, and let…

Let $L_2$ be the Beauville–Bogomolov lattice of a manifold of $K3^{[2]}$-type. Let $M$ be an admissible sublattice of $L_2$, let…

Let $C_n$ be a circle and $L_n$ a linear graph with $n$ vertices, with $C_n * L_n$ their Shannon product. The linear-graph product conjecture. … The notation is inconsistent: the l…

Let $C_n$ be a circle and $W_m$ the wheel graph with $m$ vertices, with $C_n * W_m$ their Shannon product. The wheel-product conjecture. … The formula is suggested by computations…

Let $C_n$ be a circle and $K_m$ a complete graph, with $C_n * K_m$ their Shannon product. The complete-graph product conjecture. … This is presented as an experimentally observed f…

Let $G$ be a $3$-sphere, with $f_2(S_i)$ denoting the number of $2$-faces in the unit sphere $S_i$ of a vertex and $f_0(G)$ the number of vertices of $G$. The analytic torsion form…

Let $G$ be the reductive group defining the symmetric space $\widetilde{X}$, let $\tau\in\operatorname{Rep}(G(\mathbb{R}))$ satisfy $\tau\not\cong\tau_\theta$, and let…

Let $(M,E,X)$ be the geometric data associated with the unit-cotangent bundle $M=S^*_g\Sigma$ of a closed manifold $\Sigma$, where $E$ is the flat vector bundle determined by a rep…

Let $\pi: X\rightarrow B=\operatorname{Spec}(\mathcal{O}_{K})$ be an integral, flat, projective scheme of dimension $2$, and let $L$ be a metrized line bundle on $X$. Suppose there…

Let $(\Sigma,g)$ be a negatively curved oriented compact Riemannian manifold of dimension $n>2$, and let $\alpha$ be an acyclic unitary representation of $\pi_1(\Sigma)$. Let…

Let $Y$ be a Calabi–Yau orbifold and let $\widetilde{Y}\to Y$ be a crepant resolution. Orbifold–resolution conjecture. There is a constant $C(\widetilde{Y}/Y)$, depending only on t…

General gluing conjecture. With the assumption of product structures, the following identity holds in $Q^{S}/Q^{S,0}$:

Compatibility conjecture. Under this identification of determinant lines, the Ray–Singer metric of the Gauss–Manin connection should map to the Ray–Singer metric of the original co…

Let $M$ be a fixed closed hyperbolic $3$-manifold, and let $M_n$ be covers of $M$ that Benjamini–Schramm converge to hyperbolic $3$-space $\mathbb{H}^3$. The analytic torsion conve…

Let $\pi:M\to B$ be a proper submersion between manifolds, let $V$ be a bundle of $R$-modules with geometry $h^{V}$, let $g^{\pi}$ be a Riemannian structure on $\pi$, and let…

Let $Y$ be a finite-volume hyperbolic $3$-manifold, and let Y_n_{n\in \mathbb{N}} be a nested collection of finite covers such that … is trivial. Here $\tau(Y_n,\alpha_{can})$ de…

Analytic torsion coercivity conjecture. Under these assumptions, $\mathcal{F}_{\omega_{0}}$ is coercive.

Curve critical-point conjecture. For every $k$, the functional $\mathcal{F}_{k}$ admits a critical point.

Fang's conjecture. The upper bound is achieved precisely by the Fubini–Study metric on $L$, up to scaling.

Gillet–Soulé's conjecture. The determinant of $\Delta_{\bar{\partial}}$, viewed as a functional on the space of all smooth Hermitian metrics on $L$, is bounded from above.

Let $C_\alpha S^{2p-1}_{l\sin\alpha}$ be the cone of angle $\alpha$ and length $l>0$ over the odd-dimensional sphere $S^{2p-1}$, equipped with the standard metric induced by the im…