Pure Mathematics Department, University of Waterloo
Tyrone Ghaswala
“The geometry of Yang-Mills fields, Part 03”
Throughout the spring 2013 term, we will (as a group) be reading through and lecturing on ”The Geometry of Yang-Mills Fields” by Sir Michael Atiyah. All are welcome to attend.
Zhiyou Wu
”The Newlander-Nirenberg Theorem”
The Newlander-Nirenberg Theorem states that a manifold with an almost complex structure J admits local holomorphic coordinates if and only if the eigenvector fields of J determine an involutive distribution of the complexified tangent bundle. We will explain the theorem, and show that it reduces to a local question in complex analysis of several variables. The real-analytic case can be proved using the holomorphic Frobenius theorem, which we will review. Finally we show that the smooth case can be reduced to the real analytic case via a judicious change of coordinates. The existence of such a coordinate change becomes a problem in overdetermined nonlinear elliptic PDEs.