  The results summarized here are intended as rigorous mathematical statements
on various physical models coming from condensed matter physics, statistical
mechanics (classical and quantum), quantum field theory and cold atoms physics.
The main tools are mostly those of the mathematical analyst: partial
differential equations, functional analysis, spectral theory, calculus of
variations, with some incursions into probability theory. A running thread is
the construction, by rigorous asymptotic analysis most of the time, of bridges
between different levels of mathematical modeling of physical phenomena.
  This is the manuscript for the author's habilitation thesis, summarizing
results obtained between 2011 and 2016, in collaboration with: Michele
Correggi, Mathieu Lewin, Douglas Lundholm, Phan Th\`anh Nam, Robert Seiringer,
Sylvia Serfaty and Jakob Yngvason.
