11 problems

Let $\mathsf{E}_8$ denote the root lattice in dimension $8$. A lattice is a locally optimal solution to the lattice packing-covering problem if no sufficiently small deformation im…

A perfect form is a positive definite quadratic form whose associated Delone subdivision is determined by its minimal vectors; Voronoi's first perfect form is the first form in Vor…

Reciprocal counting-function conjecture. For every $x$, one has

Weighted counting-function conjecture. For $x\geq 8$, one has

Schmutz Schaller's counting-function conjecture. For every $x$, one has

For a convex body $K\subset\mathbb R^{2}$, let $d_{2,1}(K)$ be the non-separable lattice quantity defined in the source. The Makai extremal conjecture. … with equality possible onl…

A lattice packing in $\mathbb{R}^4$ has density bounded above by the lattice three-point bound, obtained from a semidefinite-programming formulation of the three-point bound. Sharp…

Let $\code{C}^\ast$ be a formally self-dual code of length $n$. Let $f_{\code{C}}(t)$ be the polynomial associated with a formally self-dual code $\code{C}$, and let…

Let $\Gamma$ be a formally unimodular packing, and let $\Xi_{\Gamma}(\tau)$ denote its secrecy function and $\xi_{\Gamma}$ its maximum value. Formally unimodular packing secrecy fu…

Single-class HCP conjecture. All EPGMs are generated by PCs from a single equivalence class. This class has cardinality $16\ell^3$ and consists of $8$ $D$-HCP configurations and th…

Let $K$ be a hexameral domain with multi-curve $\sigma$ around its boundary. Let $t_0,t_1$ be the parameters defining the non-repeating boundary interval. Suppose that the entire n…