Math 208, Manifolds I, Fall 2024

• Lectures: TTh 1:30-3:05pm, McHenry 1279

• Text: Introduction to Smooth Manifolds by John M. Lee, Second edition, Springer 2013

• Prerequisites: point-set topology, active knowledge of basic analysis and linear algebra.

• Instructor: Viktor Ginzburg; office: McHenry 4124; email: ginzburg(at)ucsc.edu,

• Office Hours: Tu 12:45--1:25pm, Th 11:45am-12:45pm or by appointment; Location: McHenry 4124

• Tentative Syllabus: This is the first course in the geometry sequence 208-210 and 211. The main theme of the course is the notion of manifold. 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. The notion of a manifold and integration of differential forms, covered in Math 209, 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 1-5, 8, 9 of the textbook and some parts of Chapters 6, 7 and 10. A word of warning: I won't follow the textbook closely.

• Coursework: There will be weekly homework sets (not graded), one take-home midterm (40%), and a take-home final (60%).

• Homework Assignments:
  • Homework Assignment I (pdf file): Topological manifolds
  • Homework Assignment II Smooth manifolds: 1-7, 1-9, 2-2, 2-3, 2-6, 2-9, 2-10
  • Homework Assignment III (pdf file): Tangent spaces
  • Homework Assignment IV (pdf file): Vector fields
  • Homework Assignment V (pdf file): Immersions, submersions and local normal forms
  • Homework Assignment VI (pdf file): Sard’s lemma, Whitney emmbedding theorem and all that

• Take-Home Midterm (pdf file): due Thursday 11/07 in class

• Take-Home Final (pdf file): due Thursday 12/05 in class