I wish to study the paper by Melroe and Taylor: Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv in Math, Vol 55 (3), 1985, 242–315. http://www.sciencedirect.com/science/article/pii/0001870885900933 Although I have some background in pseudodifferential operators and symplectic geometry, and I can read parts of the article (phase stationary phases…) ; I have some doubts to understand the very long paper, especially the symplectic part (pages 260-268).
Would you please help me with the following:
1) what is a Whitney fold? (I searched google, wikipedia without success).
2) how they construct the involutions $J_1,J_2$ in page 261?
3) What are the illuminated, shadow region and the shadow boundary?
Thank you!