| Abstract: |
| Over recent years, many authors have sought to understand how the dynamics of the nonlinear Schr\odinger equation (NLS) arise as an \emph{effective equation}. By effective equation, we mean that solutions of the NLS equation approximate solutions to an underlying physical equation in some topology in a particular asymptotic regime. In contrast to the vast amounts of activity on the derivation of the dynamics of the NLS, to the best of our knowledge, questions about the origins of the geometric structure of the NLS have remained unexplored. In this talk, I will discuss a program joint with Mendelson, Nahmod, Pavlovi\`{c}, and Staffilani on mathematically understanding geometric properties of the NLS as a consequence of its role as an effective description of a finite system of interacting bosons in the limit as the number of particles tends to infinity. In particular, I will discuss recent work providing the first rigorous derivation of the Hamiltonian structure of the NLS, in all dimensions, from that of finitely many interacting bosons and work towards deriving the integrability of the one-dimensional cubic NLS from the exactly solvable Lieb-Liniger model. |
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