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We consider the cubic nonlinear Schrödinger equation (NLS) in any spatial dimension, which is a well-known example of an infinite-dimensional Hamiltonian system. Inspired by the knowledge that the NLS is an effective equation for a system of interacting bosons as the particle number tends to infinity, we provide a derivation of the Hamiltonian structure, which is comprised of both a Hamiltonian functional and a weak symplectic structure, for the nonlinear Schrödinger equation from quantum manybody systems. Our geometric constructions are based on a quantized version of the Poisson structure introduced by Marsden, Morrison and Weinstein [19] for a system describing the evolution of finitely many indistinguishable classical particles. ∞ 61 7. GP Hamiltonian flows 65 7.1. BBGKY Hamiltonian Flow 65 7.2. GP Hamiltonian Flow 68 1 D.M. is funded in part by NSF DMS-1800697.Appendix A. Locally convex spaces 69 A.1. Calculus on locally convex spaces 69 A.2. Smooth locally convex manifolds 70 Appendix B. Distribution-valued operators 72 B.1. Adjoint 72 B.2. Trace and partial trace 73 B.3. Contractions and the "good mapping property" 75 B.4. The subspace L gmp77 References 78Remark 1.1. We note that our work does not address any derivation of the dynamics of the nonlinear Schrödinger equation from many-body quantum systems in the vein of the aforementioned works by Erdös et al. [7,8,9]. Our current work is complementary to those in the sense that it addresses geometric aspects of the connection of the NLS with quantum many-body systems, answering questions which are of a different nature than those about the dynamics.Remark 1.2. We view this work as part of a broader program of understanding how qualitative properties of PDE arise from underlying physical problems. We also mention the works of Lewin, Nam, and Rougerie [17] and Fröhlich, Knowles, Schlein, and Sohinger [10], which derive invariant Gibbs measures for the NLS from many-body quantum systems, as we believe they are related in spirit to this program.We conclude by mentioning an application of our current work. In the one-dimensional cubic case, for which the corresponding one-dimensional cubic nonlinear Schrödinger equation is known to be integrable, we establish in a companion work [23] that there exists an infinite sequence of Poisson commuting functionals, which we call energies. The Hamiltonian flow associated to the third energy yields the GP hierarchy, and the corresponding flows for the sequence of energies yield a "hierarchy of infinite-particle hierarchies" which generalizes the Schrödinger hierarchy of Palais [29].In the next section, Section 2, we will record the precise statements of our main results, which require some additional notation and background. We postpone a subsection on the organization of our paper until the end of this next section. Statements of main results and blueprint of proofsWe will now state precisely and outline the proofs of our three main results: Theorem 2.3, Theorem 2.10, and Theorem 2.12. The first two results provide the af...

In this paper, we give a rigorous justification of the point vortex approximation to the family of modified surface quasi-geostrophic (mSQG) equations globally in time in both the inviscid and vanishing dissipative cases. This result completes the justification for the remaining range of the mSQG family unaddressed by Geldhauser and Romito [19] in the case of identically signed vortices.

A longstanding problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli [16] showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of N particles interacting in T d , d ≥ 2, via Newton's second law through a supercritical mean-field limit. Namely, the coupling constant λ in front of the pair potential, which is Coulombic, scales like N −θ for some θ ∈ (0, 1), in contrast to the usual mean-field scaling λ ∼ N −1 . Assuming θ ∈ (1 − 2 d(d+1) , 1), they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as N → ∞. Han-Kwan and Iacobelli asked if their range for θ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit N → ∞ for θ ∈ (1 − 2 d , 1). For reasons of scaling, this range appears optimal in all dimensions. Our proof is based on Serfaty's modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved "renormalized commutator" estimate to obtain the larger range for θ.