Math 209, Manifolds II, Winter 2006

• Lectures: TTh 10:00am-11:45am, JBE 360

• Text: Introduction to Smooth Manifolds by John M. Lee, Springer 2003

• Instructor: Viktor Ginzburg; office: 353C Baskin Eng.,

• Office Hours: T 12:00-1:30pm, W 2:00-3:30pm or by appointment

• Tentative Syllabus: This course is the second course in the new geometry sequence 208-210 and 211. The main theme of the course is integration on manifolds. Manifolds are curved spaces (such as the physical space-time, according to some theories) that can be thought of as a generalization of surfaces to higher dimensions. Among manifolds are Lie groups, configuration spaces of many physical systems, and in fact most of the underlying objects of modern geometry. Manifolds are treated in detail in 208. The goal of 209 is to develop a theory of integration on manifolds: what to integrate (differential forms), how to integrate, and the integral theorems (Stokes' formula). Differential forms are also omnipresent in geometry and physics. For instance, the curvature of a surface, the area or volume, and electro-magnetic fields are differential forms. The notion of a manifold and integration of differential forms are the most basic elements of the modern geometry language used in differential topology and geometry, dynamical systems, and theoretical physics (e.g., relativity and string theory). We will cover Chapters 11-16 of the textbook and some parts of Chapters 18 and 19.

• Coursework: There will be weekly homework sets, one take-home midterm, and a take-home final.

• Homework Assignments:
  • Homework Assignment I (pdf file): One-forms and integration, due Tuesday 1/17-2006
  • Homework Assignment II (pdf file): Vector bundles, due Tuesday 1/24-2006
  • Homework Assignment III (pdf file): Linear algebra, due Tuesday 1/31-2006
  • Homework Assignment IV (pdf file): Differential forms, due Tuesday 02/07-2006
  • Homework Assignment V (pdf file): More on Differential forms, due Tuesday 02/14-2006
  • Homework Assignment VI (pdf file): Orientations and integration, due Tuesday 02/28-2006
  • Homework Assignment VII (pdf file): De Rham cohomology and the Poincare lemma, due Tuesday 03/07-2006

• Take-Home Midterm (pdf file): due Tuesday 02/21-2006

• Take-Home Final (pdf file): due Thursday 03/16-2006