69 problems

For every $d$-dimensional reflexive lattice polytope $P\subset \mathbb{R}^d$, is the Ehrhart polynomial of its polar dual…

Let $P$ be a $d$-dimensional lattice simplex with the integer decomposition property: for every integer $m\ge 1$ and every $x\in mP\cap\mathbb{Z}^d$, there exist…

Let $P\subseteq\mathbb{R}^d$ be a $d$-dimensional lattice polytope with the integer decomposition property: for every integer $k\geq 1$ and every $x\in kP\cap\mathbb{Z}^d$, there e…

The conjecture asserts that there exists a polynomial $p(d,k)$ such that, for every $d,k\in\mathbb{Z}_{\ge 1}$ and every lattice polytope $P\subseteq[0,k]^d$, its monotone diameter…

For every lattice polytope $P$, the associated $q$-Ehrhart series $E_P(q,t)$ is a rational function and satisfies the corresponding $q$-reciprocity relation; moreover, the associat…

For $n\ge 2$, let $[n]=\{1,\ldots,n\}$ be a set of alternatives. For each linear ranking $\pi=(\pi(1),\ldots,\pi(n))$ of $[n]$, define its deterministic best-worst choice vector…

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For every finite preorder $\tau$, the $h$-polynomial of $\tau$ equals that of its dual preorder: $h_{\tau}(t)=h_{\tau^{*}}(t)$, where $\tau^{*}$ denotes the dual preorder.

For every finite poset $P$ with $n=|P|$, determine the facet-count gap $\operatorname{gap}(P):=f_{n-1}(C(P))-f_{n-1}(O(P))$, where $O(P)$ and $C(P)$ are respectively the order and…

For a unit vector $cvec{a}=(a^1,a^2,a^3)$ and $s oindent\text{in}cmathbb{R}$, let … be the totally umbilic planes in the upper-half-space model $(cmathbb{R}^3_+,b)$ of hyperbolic…

Norm-extension conjecture. Then $\lVert\cdot\rVert$ is the restriction to $G$ of a norm on $\mathbb Z^3$.

Vinberg's polyhedral-inequality conjecture. There exist a family of subsets $M_{i}\subset\{1,\dots,N\}$ and a family of elements $l_{i}\in\mathfrak{t}_{\mathbb{Z}}$ such that the s…

Makeev's conjecture. Every convex body $K$ in $\mathbb{R}^d$ of diameter $\leq 1$ can be covered by a translate of $R_\Delta$ for some non-degenerate simplex $\Delta$ of diameter…

Dickenstein–Nill's Cayley decomposition conjecture. If

Böröczky's conjecture. There is a point $x\in P$ with norm

Volume–Dehn separation conjecture. In Euclidean, spherical, and hyperbolic geometries, do the volume and generalized Dehn invariant separate the scissors congruence classes of poly…

Let $G$ be of type $A_n$, let $\underline{w}_0$ be any reduced decomposition of the longest Weyl-group element, and let $\lambda$ be a weight. Write $Q_{\underline{w}_0}(\lambda)$…

Let $m,n$ be positive integers, let $\xi^m_n$ denote the generator indexed by $m$ inputs and $n$ outputs, and let $\partial$ be the perturbed differential in the minimal model of t…

Polyhedral semistable reduction conjecture. There exists a projective alteration $\Delta_B^1\to\Delta_B$, with induced alteration $\Delta_X^1\to\Delta_X$, and a projective subdivis…

Neoplatonic homotopy conjecture. Every $6$-net $\sigma$ has a family of realizations $\Delta(\sigma,l)$, each unique up to isometry, as undented hyperbolic polyhedra of side length…

Volume-max conjecture. For a prime $6$-net $\sigma$, its ideal neoplatonic realization maximizes volume among all ideal geodesic $2$-cycles with combinatorics $\sigma$.

Ideal-prime conjecture. If the $6$-net $\sigma$ is prime, its ideal neoplatonic realization $\Delta(\sigma,\infty)$ is convex.

Ideal neoplatonic conjecture. Every $6$-net $\sigma$ has a realization $\Delta(\sigma,\infty)$, unique up to isometry, as an ideal neoplatonic.

Neoplatonic conjecture. Any $6$-net $\sigma$ has a realization, unique up to isometry, as an undented Euclidean polyhedron $\Delta(\sigma)$ built from equilateral triangles of side…

Let $G$ be a finitely generated group satisfying the Atiyah conjecture, and let $C_*$ be a finitely generated free $L^2$-acyclic $\mathbb{Z}G$-chain complex. For a cohomology param…