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:

Homework Assignment I (pdf file): One-forms and integration

Homework Assignment II (pdf file): Vector bundles

Homework Assignment III (pdf file): Linear algebra

Homework Assignment IV (pdf file): Differential forms

Homework Assignment V (pdf file): Orientations and integration

Exams:

Take-Home Midterm (pdf file): due Tuesday 2/27

Take-Home Final (pdf file): due Thursday 3/14