Independent sets in regular graphs

Estimating the number of independent sets of regular graphs is in the center of attention recently.
The classical result of Korshunov and Sapozhenko in 1983 counts the number of independent sets in the hypercube, and then shows that typical independent sets are not far from the trivial construction. The main focus of the talk to prove similar results for the middle two layers of the hypercube.
This is partly joint work with Bela Bollobas, Ramon I. Garcia, Lina Li, Bhargav Narayanan, Andrew Treglown, Adam Zs. Wagner.