Negative energy blowup results for the focusing Hartree hierarchy

In this talk we discuss new negative energy blowup results for the Hartree hierarchy, an infinitely coupled system of PDEs arising from the study of many-body quantum mechanics.  We obtain results both with and without an assumption of finite variance on the initial data.  The key tools involved are virial identities for the Hartree hierarchy, together with localized variants of these identities.  The most delicate case of the analysis is the proof without finite variance -- here, we use a suitable quantum de Finetti theorem and a carefully chosen truncation lemma allowing for the control of additional terms appearing from the localization procedure.