Curious why mathoverflow (se site) doesn't have hot questions list on the right. It also uses the old stackoverflow design. Why has this site's design and functionality been abandoned (or at least look abandoned?)
Math Overflow has more independence than sites that went through the A51 process. They have the right to leave the network if they choose to, and a few other minor things. They were a successful site outside the Stackexchange network, based on the SE 1.0 software. The old software wasn't supported anymore, and SE offered them to be included in the new SE 2.0 network.
They prefer not having the hot questions list, and the design is adapted from an older design created while they were still an SE 1.0 site as far as I know.
I suspect SE would create a new design if the Mathoverflow community asked for it, but they seem to be content with and used to the old design.