Closed characteristics on compact convex hypersurfaces in R8

In this research the author uses Morse theory the index iteration theory for symplectic paths developed by Y. Long and his coworkers and a new commutative property for closed characteristics to solve the n=4 case for a long standing conjecture in Hamiltonian analysis. This book is reprinted from Advances in Mathematics Vol 297 Wei Wang Closed characteristics on compact convex hypersurfaces in R8 93-148 2016 with permission from Elsevier.