Differential Topology

This subject extends the methods of calculus and linear algebra to study the topology of higher dimensional spaces. The ideas introduced are of great importance throughout mathematics, physics and engineering. This subject will cover basic material on the differential topology of manifolds. Topics include: smooth manifolds, tangent spaces, inverse and implicit function theorems; differential forms, integration on manifolds and de Rham cohomology; submersions and fibre bundles; immersions and transversality; examples coming from Lie groups and homogeneous spaces. Additional topics may include: Morse theory; intersection theory; characteristic classes and Chern-Weil theory; the Thom isomorphism; bordism theory.