Abstract: This series of lectures will cover several interconnected topics: dispersive decay of the nonlinear Schrödinger equation (NLS), NLS on waveguide manifolds, and many-body Schrödinger equations. 

The first part focuses on the dispersive decay property of NLS-namely, the fact that nonlinear solutions exhibit the same dispersive decay rates as linear ones. The presentation is primarily based on a series of joint works with Prof. C. Fan (AMSS), Prof. R. Killip (UCLA), Prof. M. Visan (UCLA), and Prof. G. Staffilani (MIT).

The second talk provides a survey of recent advances in the study of nonlinear Schrödinger equations on waveguide manifolds, covering both analytical and PDE perspectives. We will also highlight several open problems in this active research area. 

In the third talk, we will discuss many-body Schrödinger equations, focusing on key estimates and potential applications. The content is largely drawn from collaborative work with members of the UMD group, including Prof. M. Machedon, Prof. M. Grillakis, Prof. X. Huang (LSU) and Prof. J. Chong (BIT).