Paul Cusson, Department of Pure Mathematics, University of Waterloo
"A proof of the Newlander-Nirenberg Theorem"
The Newlander-Nirenberg Theorem is a result that determines exactly when an almost complex manifold is actually a complex manifold. That is, given an almost complex structure $J$ on a manifold, it is integrable if and only if a certain PDE is satisfied, namely the vanishing of the Nijenhuis tensor. We will look at a proof of this theorem given by Malgrange, which uses results from the theory of nonlinear systems of elliptic PDEs.