On the local-global conjecture for higher-dimensional Kleinian sphere packing

We will discuss the local-global conjecture for certain integral Kleinian sphere packings, such as Soddy sphere packings and orthoplicial Apollonian sphere packings. (These sphere packings are 3-dimensional analogues of Apollonian circle packings.) Sometimes each sphere in a Kleinian sphere packing has a bend (1/radius) that is an integer. When all the bends are integral, which integers appear as bends? In 2019, Kontorovich proved the local-global conjecture for Soddy sphere packings. Work towards proving a local-global conjecture for orthoplicial Apollonian sphere packings has been done by Dias and Nakamura. We will discuss work concerning local-global conjectures for Kleinian sphere packings.