Explore the intricacies of nonlinear Schrodinger equations and their periodic solutions in this engaging SIAM PDE seminar. Delve into a diverse array of mathematical techniques spanning harmonic and Fourier analysis, dynamical systems, number theory, and probability. Begin with the derivation of these equations from many-body systems and examine how Hamiltonian structures are preserved through this process. Investigate the long-term dynamics of associated initial value problems, focusing on energy transfer concepts. Discover how dynamical systems theory is crucial for developing even basic statements, and learn about recent advancements in rigorously deriving wave kinetic equations for multidimensional KdV-type equations using tools such as Feynman diagrams, sharp dispersive estimates, and improved combinatorial lemmata.