Analys på Mångfalder (10hp), VT 2011

John Milnor has been awarded the Abel prize "for pioneering discoveries in topology, geometry and algebra". See the official announcement (also in English), with further links, including The work of John Milnor.

Schema        This week

Kursstart 17 januari 15.15.

Kurskod 1MA037.

Kursplan.        Studentportalen

Lärare

Kurslitteratur

See also §3 of my lecture notes (for a different course with emphasis on Riemannian geometry):

En kort engelsk-svensk ordlista finns nedan.

Homework

Three homework assignments will be given. These form part of the examination, and should be solved individually. Correct solutions will give bonus points on the exam in May. (Up to 3 bonus points per set, i.e., total maximum 9.) Solutions can be given to me at lectures, or in my mailbox at Ångström (floor 4), at the latest on the days shown below.
  1. 2 March.
  2. 6 April.
  3. 16 May.

Lectures

Here is a rough list of contents for the lectures, with approximate indications of sections in the textbooks (M = Milnor; BT = Bott and Tu).
Recommended exercises are often given. They will be solved at a later lecture (marked Problem solving).
  1. 17/1: Differentiable submanifolds of RN. Charts. Diffeomorphisms. Inverse function theorem. M §1. Exerc.
  2. 24/1: General differentiable manifolds. Charts. Tangent vectors, tangent space; Differential. M §1
  3. 26/1: Tangent bundle. Immersion. Submersion. Regular point, singular point. Regular value, singular value. M §1. Exercises M 8:8-10.
  4. 28/1: Oriented manifolds. Manifolds with boundary. Sard's theorem. M §2-3.
  5. 31/1: Degree of a smooth map. Degree modulo 2. Homotopy. Isotopy. M §4-5. Exercises M 8:1-3, 6.
  6. 8/2: Vector field. Index. M §6. Exercises M 8:11-12.
  7. 15/2: Framed cobordism. Pontryagin construction. Exercises M 8:13.
  8. 2/3: PROBLEM SOLVING. M §8.1-13
  9. 10/3: Differential forms. BT §1.1
  10. 16/3: Exterior derivative. BT §1.1; Example 1.6, Exercise 1.7
  11. 21/3: De Rham cohomology. Categories and functors. Long exact sequence. Mayer-Vietoris sequence. BT §1.2
  12. 23/3: Mayer-Vietoris for compact supports. The Poincaré lemma. BT §1.2, 1.4; Exercise 4.8
  13. 28/3: Partitions of unity. Integration. Stoke's theorem. BT §1.3, 1.4; Exercises 4.2, 4.3, 4.3.1.
  14. 30/3: PROBLEM SOLVING. Poincaré lemma with compact support. BT §1.4; Exercise 4.5
  15. 4/4: Poincaré duality. Good covers. BT §1.5; Exercise 5.5
  16. 6/4: Degree of maps. Poincaré duals of a submanifold. Künneth formulas. BT §1.4, 1.5; Exercise 4.10.1, 5.12, 5.15, 5.16 and: Find the cohomology of the torus.
  17. 11/4: PROBLEM SOLVING. Euler-Poincaré characteristic.
  18. 2/5: Vector bundles. BT §1.6. Exercise 6.2, 6.10.
  19. 4/5: Thom isomorphism. Thom class. Exercise 6.20
  20. 9/5: Euler class. Exercise 6.32, 6.36, 6.43, 6.44, 6.45, 6.46.
  21. 16/5: PROBLEM SOLVING. Exam 17/4 2009.
  22. 18/5: PROBLEM SOLVING.

Exam

Check exam schedule for changes.

Old exams

En kort ordlista

Detta är ett litet komplement till Anders Vretblads allmänna matematiska ordlista.
Engelska Svenska
chart karta
differential form differentialform
exterior yttre
manifold mångfald
smooth glatt