Differential Forms in Topology

A brief history of the subject:

Differential forms play a predominant role in many areas of mathematics like topology , geometry , analysis , and in modern theoretical physics . They have the unique feature that they stand at a crossroad of analytic and algebraic methods to probe topological invariants of a space.

This course is an introduction to applications of differential forms in topology and as such should be of interest to graduate students ( as well as enthusiastic undergraduates and faculty members ) in Mathematics , Applied Mathematics and Elementary Particle Physics . The lectures will be self -contained and I will cover the necessary background as they may be required.

Course Outline:

• de Rham theory ( de Rham complex , Poincaré lemma , Mayer-Vietories argument ,Poincare duality , Thom isomorphism , Euler and Thom class, Cech-de Rham complex deRham's theorem )
• characteristic classes ( Chern-Weil theory , Chern and Pontrjagin classes , universal bundles )
• time permitting,Hodge theory (harmonic forms, Hodge decomposition theorem ) .

Marking Policy:  Students are expected to prerare and present reports on selected topics of current interest . I will choose these topics in consultation with each student .

Textbook:

Differential Forms in Algebraic Topology , by R. Bott and W. Tu , published by Springer-Verlag.