Low-density materials surrounded by a thin shell are still going to weight a lot more than air surrounded by a thin shell. Also, you have to look into how this

Low-density materials surrounded by a thin shell are still going to weight a lot more than air surrounded by a thin shell. Also, you have to look into how this is going to be heated. For a hot air balloon to work, all the air inside has to be hot. If your balloon is filled with an amazingly efficient insulator.... Also, the reason airships use lighter-than-air gas is because then they don't have to carry lots of heavy fuel for heating air. Things like fuel, and pipes to carry heat throughout your aerogel, will rapidly eat up any efficiencies you gained.