Lecturer: Dmitry Tonkonog.

Links: University course homepage, brief syllabus.

Homework 1. |
Worth 20 points, | Released on 10 Feb, | Due by 24 Feb, 01:00pm |

Homework 2. |
Worth 20 points, | Released on 8 March, | Due by 29 March, 01:00pm |

Homework 3. |
Worth 30 points, | Released on 13 April, | Due by 4 May, 01:00pm |

[L01] | [L02] | [L03] | [L04] | [ExS] | [L06] | [L07] | [L08] | [L09] | [ExS] | [L11] | [L12] |

[L13] | [L14] | [ExS] | [L16] | [L17] | [L18] | [L19] | [L20] | [ExS] | [L22] | [L23] |

NB: the notes have typos.

These exercises are ungraded, and are discussed during problem sessions:

[Ex 01-02] | [Ex 03-04] | [Ex 06-07] | [Ex 08-09] | [Ex 11-12] | [Ex 13-14] | [Ex 16-18] | [Ex 19-20] |

Schedule of problem sessions:

Lec 5, |
31 Jan |

Lec 10, |
17 Feb |

Lec 15, |
8 March |

Lec 21, |
13 April |

1. Excursion: The pendulum. Smooth maps and diffeomorphisms.

2. Manifolds, tangent planes, differential of a smooth map.

3. More on the differential. Inverse function theorem. Submersions.

4. Proof that SO(n) is a manifold. Immersions. Irrational winding on the torus and other pathological examples.

5. Exercise session.

6. Maps transverse to submanifolds. Preimage theorem. Transversally intersecting submanifolds. Sard's theorem: statement. Measure 0 sets.

7. Proof of Sard's theorem.

8. Morse functions. Excursion: Morse homology.

9. Existence of Morse functions. Quick excursion: Vector bundles. Whitney embedding theorem.

10. Exercise session.

11. Manifolds with boundary. Preimage theorem for maps from manifolds with boundary. Non-existence of retractions to the boundary.

12. Brouwer fixed point theorem. Degree modulo 2.

13. Degree and the fundamental theorem of algebra. Transversality theorem. Epsilon-neighbourhood theorem and the normal bundle.

14. Transversality theorems continued. Intersection theory mod 2. Examples for surfaces.

15. Exercise session.

16. Sketch of the definition & invariance of the integral degree. Orientations.

17. Orientations, continued. Oriented intersection theory.

18. Euler characteristic. Lefschetz fixed point theory.

19. Vector fields and the Poincare-Hopf theorem.

20. Homotopy groups. Hopf degree theorem. Pontryagin theory.

21. Exercise session.

22. Pontryagin theory, continued. Fundamental groups of surfaces, Dehn's solution of the word problem.

23. Excursion: mapping class groups, Nielsen-Thurston classification, sketch proof of the Nielsen realisation theorem.

The final grade is determined by three compulsory graded homework assignments. The total number of points one can earn from the three homework assignments is 70, and the final grades are determined according to the table:

Grade 5: |
56-70 points |

Grade 4: |
46-55 points |

Grade 3: |
32-45 points |

Additionally, you must earn at least 25% of points for each of the three homeworks in order to get Grade 3, 4 or 5.