MA721
This is a first year graduate class in differential topology. A syllabus can be found here.
Time: M/W/F 12:20 PM -- 1:10 PM.
Location: WED 212

Course schedule and notes

In parentheses are the (approximate) sections of the textbook.
  • September 6: Topological manifolds (1.1). Notes.
  • September 8: Derivatives and the local definition of smoothness (1.2). Notes.
  • September 11: The definition of a smooth manifold (1.2-1.3). Notes.
  • September 13: Examples of smooth manifolds, smooth functions (2.1). (Note the extended example of the Grassmannian manifold which was not covered in class). Notes.
  • September 15: Smooth maps and diffeomorphisms (2.1). Notes.
  • September 18: The tangent space at a point in Euclidean space (3.1). Notes.
  • September 20: The tangent space at a point in a smooth manifold (3.1-3.2). Notes.
  • September 22: The differential and the tangent bundle (3.2-3.4). Notes.
  • September 25: Submersions and immersions (4.1). Notes.
  • September 27: Constant rank theorem (4.1). Notes.
  • September 29: Embeddings (4.2). Notes.
  • October 2: Submanifolds (5.1). Notes.
  • October 4: Level sets as submanifolds (5.1). Notes.
  • October 6: No class.
  • October 10 (note the BU Monday): Vector fields, basic properties (8.1). Notes.
  • October 11: Vector fields as derivations, pushforward (8.1-8.2). Notes.
  • October 13: Vector fields along submanifolds (8.2-8.3). Notes.
  • October 16: Lie groups, basic concepts (7.1-7.2). Notes.
  • October 18: Lie algebras, the Jacobi bracket (7.2,8.3). Notes.
  • October 20: The Lie algebra of a Lie group (8.4). Notes.
  • October 23: Integral curves and flows (9.1-9.2). Notes.
  • October 25: Examples of flows (9.2). Notes.
  • October 27: The Lie derivative (9.5) Notes.
  • October 30: Vector bundles; basic definitions (10.1). Notes.
  • November 1: Vector bundles; examples and gluing (10.1). Notes.
  • November 3: Review.
  • November 6: Vector bundles; gluing data (10.2). Notes.
  • November 8: Vector bundles; frames (10.2-10.3). Notes.
  • November 10: Vector bundles; categorical aspects (10.3-10.4). Notes.
  • November 13: Dual vector space, covectors (11.1). (No class notes)
  • November 15: Cotangent bundle, (11.1-11.2). Notes.
  • November 17: The exterior derivative, pullbacks of one-forms (11.2).
  • November 20: Tensor products, exterior products (12.1-12.2).
  • November 27: More exterior products, the top exterior product and determinants (12.2).
  • November 29: Differential forms, pullbacks. (14.1-14.2).

    • Homework

  • Homework 1, due September 15. Solutions.
  • Homework 2, due September 22 (edited on September 20). Solutions.
  • Homework 3, due September 29. Solutions.
  • Homework 4, NOW DUE October 10. Solutions.
  • Homework 5, due October 13. Solutions.
  • Homework 6, NOW DUE October 23. Solutions.
  • Homework 7, NOW DUE November 3.
  • Practice. Not to be turned in.
  • Homework 8, due November 10. Solutions.
  • Homework 9, due November 21. Solutions.
  • Homework 10, not to be turned in.