General information
Course description
This course is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. I will largely follow the standard syllabus. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle, and a section of the tangent bundle is a vector field. Alternatively, vector fields can be viewed as first-order differential operators. We will study flows of vector fields and prove the Frobenius integrability theorem. In the presence of a Riemannian metric, the notions of parallel transport, curvature, and geodesics are developed. We will examine the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics will be covered.
This is one of the core courses in the Cornell PhD program in Mathematics, and can be used to satisfy the core course requirement in the program. The course is also suitable for students in many other Cornell PhD programs, such as Physics or the Center for Applied Mathematics.
Prerequisites
Undergraduate analysis, linear algebra, and point-set topology.
Homework
There will be homework due almost every week, a portion of which will be graded and returned in class a week later. Your grade will be determined by your performance on the homework.