微分形式在代數拓樸的應用

The first goal of this course is to study the theory of De Rham Cohomology for finite dimensional smooth manifolds. We will develop techniques for the computations of De Rham Cohomology groups of smooth manifolds. We will prove the Poincare duality theorem in the framework of De Rham Cohomology. In general, we will employ differential forms as the main tool for our study here. So, in some sense, the focus will be on the differential-topological aspect of the study of the topology of smooth manifolds. Next, we will also study the topology of vector bundles over smooth manifolds. The main references for this course will be Chapter 10 of the first volume of the classical textbook Differential geometry by Spivak, as well as Chapter 13 of Volume 5 of Differential Geometry by Spivak.

Basic training in analysis, linear algebra, and undergraduate level topology.

attendance (25precent), notes taking (30 percent ), and a Final Project (7 percent), Quiz (8 precent), homework (15 precent) a take home exam (15 percent).