4 problems

Let $\Phi_4$ be the set of all $\mathbb{Z}$-lattices of rank $4$. Write $I_4$ for the rank-$4$ identity lattice, and let $A_4$ and $A_2$ denote the lattices used in the criterion s…

Let $\ell_1$ and $\ell_2$ be binary $\mathbb{Z}$-lattices with equal discriminants … Assume that the pair $(\ell_1,\ell_2)$ is buried in a genus of rank $3$. Burying conjecture. Th…

Let $\Lambda$ be an irreducible integral lattice generated by vectors of norm $3$. A lattice is $\sigma$-integrable if its bilinear form, after multiplication by $\sigma$, admits a…

Let $G$ be a connected graph, and let its smallest eigenvalue be denoted by $4\lambda_{\min}(G)4$. A graph is $\sigma$-integrable if its associated integral lattice admits an embed…