Defying gravity with vertical shaking? I saw  this video on levitating a denser liquid on less dense air by vertical shaking. But I couldn't understand some points in the video. Particularly:
How did the liquid levitate?
I think it's because of vibrating the air in the vertical direction, which causes it to apply a force on the liquid in the upward direction.
But how does vertical shaking affects the density of the liquid and makes it rise on a less dense air  ?
 A: 
How did the liquid levitate at 0:37 or at any one of those moments ?

The liquid levitates because the gauge pressure of the gas pushing up on the bottom surface of the liquid is equal to the weight of the fluid on average. This is essentially the same reason a hovercraft levitates.

I think it's because of vibrating the air in the vertical direction which causes it to apply a force on the liquid in the upward direction.

The purpose of the vibrations is simply to prevent droplets from forming. By preventing drops from forming the air cannot get around the liquid and let it fall. The shaking is what keeps the “hovercraft” intact.

Is there any mathematical model of this force so that we can find the exact frequency for this levitation ?

Yes, the full derivation of this phenomenon is here.
https://arxiv.org/abs/2003.04777
Note, in the derivation they find that inside the fluid is a location where gas bubbles above that line float up and gas bubbles below that line float down. As the lower bubbles float down they are compressed and get smaller. This gives them the requisite pressure to hold up the liquid after they leave the lower surface.
A: I guess this situation is similar to the following question.
https://physics.stackexchange.com/a/615688/289569
I think the first case in my answer(where the space between the puck and wall of the container is waterproof) somewhat explains this phenomenon.
A: Well that was a cool video!  There are three things operating at the same time to make it work...the pressure of the air, the surface tension of the liquid and the energy from the vibrator.  The air pressure supports the liquid.  The surface tension of the liquid holds the liquid together and provides the seal for the pressurized air.  The vibrator provides vibratory energy (motion) to stabilize the system, otherwise the liquid would ooze to the bottom like syrup under the force of gravity.  That would be fun to try out in zero gravity on the space station!
A: *

*when you apply vibrations and then flip the liquid then due to vibrations no droplets are formed and high surface tension of the liquid doesn't allow mixing of anything in it easily
so some particular density may allow that to happen but some completely block the mixing of the air

