Incidence geometry and the cubic NLS

par Prof. Sebastian Herr (Universität Bielefeld)

lundi 11 mars 2024, 13:00 → 15:15 Europe/Paris

435 (UMPA, ENS de Lyon)

We prove the optimal L⁴-Strichartz estimate for the Schrödinger equation
on the two-dimensional rational torus, which improves an estimate of Bourgain. Instead of analytic number theory we employ the Szemeredi–Trotter Theorem, which gives an upper bound on the number of lines m passing through at least k points of a given set of n points in the plane. This new approach yields an even stronger L⁴ bound on a logarithmic time scale, which implies
global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger
equation in H^s for any s > 0 and data which is small in the critical L² norm. This is joint work with Beomjong Kwak (KAIST).