238 problems

Let $G=\mathrm{PSL}_2(\mathbb{R})^d$ and let $\Gamma$ be a lattice in $G$. For a bounded family $\mathfrak{B}\subseteq\widehat{G}$ of non-tempered irreducible unitary representatio…

Let $f:({\mathbb C}^{m+1},0)\to({\mathbb C},0)$ be an isolated quasihomogeneous hypersurface singularity, let $(H_{\mathbb Z},M)$ be its associated integral lattice with monodromy,…

Let an $n$-dimensional quantizer be a quantization scheme in $\mathbb{R}^n$, and distinguish lattice quantizers from arbitrary quantizers. Gersho's conjecture. For all sufficiently…

Let $C$ denote the $n$-dimensional unit cube, let $k$ be a positive integer, and let $\Lambda$ be an $n$-dimensional lattice. A $k$-fold lattice tiling $C+\Lambda$ is a $k$-fold tr…

Let $g(k)$ be the maximum cardinality of a finite set $X\subset\mathbb{R}^2$ determining at most $k$ distinct distances, and call a set attaining this maximum an extremizer. Erdős–…

Let $H$ be a higher-rank semisimple group of the form … where each $F_i$ is a local field, each $\mathbf{G}_i$ is an absolutely almost simple $F_i$-group, and … For a group…

Let $X_d$ be the space of lattices, let $\mathcal{WR}_d$ denote the set of well-rounded lattices, and consider the covering-radius condition from the source: for every…

Equidistribution conjecture. The equidistribution result known for cubic, quartic, and quintic number fields should also hold for number fields of degree $n>5$.

Margulis' conjecture. There exists a neighborhood $U\subset G(\R)$ of the identity such that for any irreducible arithmetic cocompact lattice $\Gamma\subset G(\R)$, the intersectio…

Let $L\subset {\mathbb R}^n$ be a unimodular lattice, and define the multiplicative norm of ${\boldsymbol x}=(x_1,\ldots,x_n)$ by $N({\boldsymbol x})=|x_1\cdots x_n|$. Minkowski's…

Margulis–Zimmer conjecture. Every commensurated subgroup of $\Lambda$ is either finite or $S'$-arithmetic for some $S'\subseteq S$.

Let $G=\prod_{i=1}^m G_i(k_i)$, where the $k_i$ are locally compact, non-discrete fields and the $G_i$ are connected, absolutely almost simple algebraic groups defined over $k_i$.…

Consider packings of unit balls in $\mathbb{R}^4$, and write $\Delta_4$ for the maximal packing density, $B_4$ for the volume of a unit ball, and $\delta_4=\Delta_4/B_4$ for the ma…

Let $X_d$ be the space of lattices under consideration, let $A$ be the group acting on $X_d$, and let $\mu=\operatorname{CovRad}_N$ be the covering radius associated with…

Let $\Gamma$ be any lattice in $\mathop{\mathrm{SO}}\nolimits(1,n)$. Thurston's conjecture. There are a finite-index subgroup $\Gamma'$ of $\Gamma$ and a surjective homomorphism ……

Let $X$ be an irreducible Euclidean building of dimension $\geq 2$, and let $\Gamma<\operatorname{Aut}(X)$ act properly and cocompactly on $X$. The group $\Gamma$ is virtually simp…

Weak-order lattice conjecture. The weak order of $(G,F)$ always has the structure of a lattice.

Let $G$ be a connected semi-simple Lie group with no compact simple factor, and suppose that the Lie algebra of $G$ has $\mathbb{R}$-rank at least $2$. Let $\Gamma$ be an irreducib…

A lattice $L\subset\mathbb{R}^{48}$ determines a lattice sphere packing by placing congruent balls at its lattice points; its density is maximized when the ratio of the ball radius…

Invariant-lattice classification conjecture. Every primitive, $\operatorname{GL}_3(\mathcal{O}_K)$-invariant lattice $L$ in $\mathcal{V}(\mathcal{O}_K)$ is…

Non-biautomaticity conjecture. The group $\Gamma$ is not biautomatic.

Regular simplex lattice packing-minima conjecture. For every $1\leq j\leq n$,

Non-orderability conjecture for higher-rank lattices. Lattices in simple Lie groups of rank at least two have no left invariant total order.

Let ${\mathcal{L}_n}$ denote the space of unimodular lattices in ${\mathbb{R}}^n$, and let $\kappa_n$ be the infimum of their Mordell constants, where the Mordell constant of a lat…

A configuration $\mathcal P$ is universally optimal if it minimizes energy for every sufficiently rapidly decreasing potential $\varphi$ that is completely monotonic as a function…