Math 209,
Manifolds II, Winter 2024
- Lectures: TTh 9:50-11:25, McHenry 1279
- Text: Introduction to Smooth Manifolds
by John M. Lee, Springer 2013 (Second Edition)
- Instructor: Viktor Ginzburg; office: McHenry 4124.
email: ginzburg(at)ucsc.edu
- Office Hours: Tu 11:30am-12:30 or by appointment
- Tentative Syllabus: This course is the second course in the
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 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, mirror symmetry, and string theory). We will cover Chapters
10-16 of the textbook and some parts of Chapters 17 and 18.
- Coursework: There will be homework sets (not graded), one take-home
midterm (40%), and a take-home final (60%).
- Homework Assignments and Exams: