Alvaro Carballeira will give a talk on 09/29/2026.

A principle in nonlinear Schrödinger (NLS) equations has been that defocusing mechanisms promote dispersion and prevent singularities from forming. In this talk, I will discuss joint work with Kevin Buck, Javier Gómez-Serrano, and Jia Shi where we proved that energy supercritical defocusing NLS equations admit asymptotically self-similar solutions that lead to blow-up in finite time. Thus, we disprove global well-posedness in dimensions 3 and 4, closing in these cases a longstanding conjecture of Bourgain. The talk will present the result and then focus on the construction of the self-similar profile, which relies on a computer-assisted shooting argument.