20 problems

Let $P$ be a flexible polyhedron in one of the constant-curvature spaces $\mathbb{E}^n$, $\mathbb{S}^n$, or $\mathbbl{\Lambda}^n$. Two polyhedra are scissors congruent if they can…

Goncharov's quadric conjecture. The map $\phi^q$ is an isomorphism.

Let $D_2$ be the Bloch–Wigner dilogarithm, and consider its values $D_2(z)$ at algebraic arguments $z$. Zagier's conjecture. Every rational linear relation among these values is a…

Let $M$ be a weakly-gentle cusped manifold, and let $H_N(M)$ be its quantum dilogarithmic invariant. Let $\mathfrak c_{\mathcal I_{fc}}(M)$ denote its ${\mathcal I}_{fc}$-scissors…

Let spherical polyhedra in $cmathbb{S}^3$ be considered up to scissors congruence, and let their volume and Dehn invariant be the corresponding invariants. Spherical scissors-congr…

Let $\widetilde{\mathcal{P}}(S^{n-1})$ be the reduced spherical scissors congruence group, and let the direct sum … be equipped with the Sah algebra Hopf structure, whose coproduct…

Let $G$ be a finite group. The equivariant $SK^G$-groups, as $H$ varies over the subgroups of $G$, form a Mackey functor, and let $K_G(\underline{\mathbb{Z}})$ denote the genuine…

Let $G$ be a finite group, and consider unoriented $G$-manifolds and their slice type Euler characteristics. Kosniowski's conjecture. The slice type Euler characteristics are a com…

Let $E^n$ be the $n$-dimensional Euclidean space and let $P\subseteq E^n$ be a polytope. The abelianisation of a group is its quotient by its commutator subgroup. Zakharevich's con…

Negative super-concrete multitiling conjecture. For every $d\geq 3$, not every super concrete lattice polytope $P\subset\mathbb R^d$ can multitile the space; equivalently, there ex…

Let $P_*(S^d)$ denote Goncharov's Dehn complex, and let $P_*(S^{2n-1})$ be its spherical version. Write $gr^gamma_n K_{n+m}(C)_Q$ for the $n$-th graded piece of the $gamma$-filtrat…

Weak Strong Bellows Conjecture. The Dehn invariant of any flexible polyhedron remains constant during the flexion. This is a weaker form of the Strong Bellows Conjecture, since sci…

Let $M_n$ and $M_I$, for $I\in\{0,1,2\}^{n+1}$, be the hyperbolic manifolds described in the paper. Since they decompose along totally geodesic surfaces into isometric pieces, they…

Let $M$ be a hyperbolic manifold, and let its stable scissors congruence class mean its scissors congruence class after adjoining a common hyperbolic polyhedron. Let…

For an angle $\theta$, let $\operatorname{L}(\theta)$ denote the Lobachevsky function. Consider angles $\theta$ that are rational multiples of $\pi$, and rational coefficients…

Strong Bellows Conjecture. If an embedded polyhedron $P_1$ is obtained from an embedded polyhedron $P_0$ by a continuous flex, then $P_1$ and $P_0$ are scissors congruent: there ex…

Bloch–Goncharov hyperplane conjecture. The map $\phi^h$ is an isomorphism.

Let $P\subset\mathbb{R}^n$ be a rational polytope with minimum Ehrhart quasi-period $N$. Two subsets of $\mathbb{R}^n$ are $\operatorname{GL}_n(\mathbb{Z})$-equidecomposable if the…

Let $P\subset\mathbb{R}^n$ be a rational polytope, and let its Ehrhart quasi-polynomial have quasi-period $1$. An open simplex is the relative interior of a simplex, and it is inte…

A rational polytope is a polytope whose vertices have rational coordinates. A subdivision decomposes a polytope into polyhedral pieces, and affine unimodular transformations are af…