In this work we consider a generalization of the symmetry classification of topological insulators to non-Hermitian Hamiltonians which satisfy a combined P T -symmetry (parity and time-reversal). We show via examples, and explicit bulk and boundary state proofs that the typical paradigm of forming topological insulator states from Dirac Hamiltonians is not compatible with the construction of non-Hermitian P T -symmetric Hamiltonians. The topological insulator states are P T -breaking phases and have energy spectra which are complex (not real) and thus are not consistent quantum theories.

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain's level set result on Strichartz estimates associated with Schrödinger equations on torus. Some sharp estimates on L 2(d+2) d norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on L 2 (Z). It was conjectured by Bourgain inThe understanding of this conjecture is still incomplete. For instance, the desired upper bounds for A 5,1,N , A 3,2,N or A 2(d+2) d ,d,N for d ≥ 3 are not yet obtained. The most crucial estimate established by Bougain in [1] is certain (sharp) level set estimate. In this paper we provide a different proof of the level set estimate.These problems arise from the study of periodic nonlinear Schrödinger equations:(1.4) ∆ x u + i∂ t u + u|u| p−2 = 0 u(x, 0) = u 0 (x) .

In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear term F (u)∂xu provided F ∈ C 5 and the initial data φ ∈ H s with s > 1/2.

We consider the blowup rate for blowup solutions to L 2 -critical, focusing NLS with a harmonic potential and a rotation term. Under a suitable spectral condition we prove that there holds the "log-log law" when the initial data is slightly above the ground state. We also construct minimal mass blowup solutions near the ground state level with distinct blowup rates.