This volume is composed of invited expository articles by well-known mathematicians in differential geometry and mathematical physics. They have been arranged in celebration of Hideki Omori's recent retirement from Tokyo University of Science and in honor of his fundamental contributions to these areas. The papers all focus on recent trends and future directions in symplectic and Poisson geometry, global analysis, infinite-dimensional Lie group theory, quantizations and noncommutative geometry, as well as applications of PDEs and variational methods to geometry. These articles will appeal to graduate students in mathematics and quantum mechanics, as well as researchers, differential geometers, and mathematical physicists.