Abstract: Asymptotes of hyperbolas were studied by Apollonius around 200 BC, and can be understood as shapes enjoying the symmetry of a 1-dimensional Lie group. Consider a Lie group $G$ acting linearly on a vector space $V$. We say $v\in V$ is nilpotent if $(\mathbb{R}^{\times})^2\cdot v\subset G.v$; when the action is matrix conjugation the notion agrees with that of nilpotent matrices. We define the asymptotic cone of an orbit as:

$Cone(G.v):=\{n\in V\;|\;n\text{ is nilpotent, }n\in\overline{(\mathbb{R}^{\times})^2\cdot G.v}\}$.

This generalizes classical asymptotes. When $V$ is the (dual) Lie algebra of a typical Lie group $G$ such as the orthogonal group, the asymptotic cone $\mathrm{Cone}(G.v)$ is completely understood. The analogous question is however open when we replace $\mathbb{R}$ by a p-adic field. In this talk, after reviewing these constructions, we motivate why they are interesting to representation theory and number theory, and explain the challenge in the "adic" setting.