12 problems

Let $\Lambda$ be a Leech lattice in phase space of dimension $24$, with arbitrary fixed density greater than $1$. The Leech-lattice conjecture. There exists a Leech lattice that so…

Let $\Lambda$ be a lattice in $\mathbb{R}^{24}$ and let $g$ be the Schwartz function constructed by the paper's linear-programming method, with $g(x)\leq 0$ beyond the relevant vec…

Assume that all nonzero vectors of a lattice $\Lambda\subset\mathbb{R}^{24}$ have length exactly $2$ or at least $\sqrt{6}$. Let $g\colon\mathbb{R}^{24}\to\mathbb{R}$ be a Schwartz…

Let $S_i$ be a subset of shortest Leech lattice vectors whose pairwise cosine values are bounded by $1/4$, with $|S_i|=496$, and let the associated construction define a 25-dimensi…

Let $T_{[abc]}$ be a lattice in the notation of the paper, and let $W$ be a frame lattice corresponding to $T_{[abc]}$. A root is a lattice vector of minimal norm as defined in the…

A tight projective $5$-design in $\mathbb{OP}^2$ is the unique system of $819$ points corresponding to the lines of the generalized hexagon $\mathrm{GH}(2,8)$, and it exhibits…

Let $\Lambda$ be the Leech lattice, and let $g$ range over conjugacy classes of its automorphism group with non-trivial fixed-point lattice. Borcherds's conjecture. Each such conju…

For each integer $g$ with $0\leq g\leq 10$, let $\Theta_{12}^g$ be the subspace of Siegel modular forms of weight $12$ and degree $g$ generated by theta series of even unimodular l…

Let $L$ be a lattice and consider $2U\oplus L(-1)$. Assume that $2U\oplus L(-1)$ has a modular form with complete $2$-divisor and that $L$ has no $2$-root. Complete 2-divisor conje…

Let $L$ be a positive-definite lattice such that $2U\oplus L(-1)$ is a $2$-reflective lattice of type (b), meaning that every lattice in the genus of $L$ has no $2$-root. Leech-lat…

Let $\Lambda$ be the Leech lattice, let $O(\Lambda)$ be its isometry group, and let $\Lambda_g$ denote the sublattice orthogonal to the fixed-point sublattice of $g\in O(\Lambda)$.…

E8 and Leech limiting-ratio conjecture. In the $E_8$ case,