6 problems

Let $d$ be a positive integer, and let the universal lattices be the groups $\mathrm{SL}_d(\mathbb{Z}[x_1,\dots,x_k])$. Shalom's conjecture. Many properties of the groups…

Consider the $34$ ternary lattices listed in the theorem immediately preceding the conjecture, over the relevant real quadratic fields. A universal ternary lattice is a ternary qua…

Let $D$ be a positive squarefree integer, let … and let $\mathcal{O}_F$ denote the ring of integers of $F$. A universal ternary $\mathcal{O}_F$-lattice is a universal quadratic lat…

Let $F$ be a number field, and let a classical universal ternary lattice mean a universal quadratic lattice of rank $3$ with the classical integrality property. Strong Kitaoka's co…

Nonexistence conjecture. If $D>56$ and $F$ has class number $1$, then there is no totally real quartic field $K$ containing $F$ such that an $\mathcal{O}_F$-lattice is universal ov…

Finite universality conjecture. The monoid variety generated by the monoids in $\mathbb{H}^1$ is finitely universal, meaning that every finite lattice is order-embeddable into its…