Manifolds (Differentieerbare varieteiten)
Blok 1, 2024/2025 (UNDER CONSTRUCTION)

This is the web-site for the course "Differentieerbare varieteiten" given in blok 1, Fall 2024. We will try to keep here all the practical informations about the course, updates/changes that take place during the year, lecture notes, etc. The most recent additions are:

Lecture notes 2024/2025 (a rather substantial revision of the old lecture notes). I am trying to arrange that paper copies will be available with the A–Eskwadraat.

THE LECTURES:

Mondays: 11:00-12:45, in BBG - 083.

Wednesdays: 11:00-12:45, in BBG - 205.

LECTURER: Marius Crainic.

THE WERKCOLLEGES (devoted to solving exercises from the lecture notes, getting feedback for the homeworks, and/or asking for help/discussing aspects of the course that you did not understand):

Mondays: 09.00 - 10.45 in BBG - 083.

Wednesdays: 09.00 - 10.4 in BBG - 205.

TEACHING ASSISTANTS: Anna Fokma and Sven Holtrop.

HOMEWORKS: TBA

Each homework will be corrected taking into account also the mathematical quality of the writing, which will count for 25 percent of the mark. If you score low on that, you have the chance to improve your writing (hence also your mark) by submitting your homework a second time (but please do not abuse, otherwise we have to put an upper limit on how many homeworks each one of you can resubmit).

Passing the course: that will be subject to some minimal requirements, and the final marks will depend on several items:

Hand in exercises- There will be mandatory weekly homework exercises. These lead to an average grade H with 1 decimal of accuracy (maximum 10).

Final exam: At the end of the course there will be a 3 hour written exam, leading to a grade E with 1 decimal of accuracy (maximum 10).

Bonus exercises: there may be some bonus exercises, very difficult ones, which would allow you to earn "bonus points" $B$.

Final mark/minimal requirement: The final grade F is be determined by
F= min{10, max{(4 E+ 6 HW)/10, (8E+ 2H)/10}+ B}
rounded off to an integral number up to 6, and to a half integer above 6 (to the closest one). The requirements for passing the exam are: H and E have to be at least 5 (before rounding!) and F has to be at least 6. Please be aware that there will be no automatic rounding up for marks below 5 (e.g. 4.95 will not be a passing mark but, in such borderline cases (if any).
Retake: the same rules and the same formula apply for the retake.

LECTURE NOTES: Lecture notes 2024/2025 (a rather substantial revision of the old lecture notes). I am trying to arrange that paper copies will be available with the A–Eskwadraat..

THE SCHEDULE WEEK BY WEEK (I will try to add, after each lecture, a description of what was discussed in the lectures + the exercises from the lecture notes that you are supposed to do during the werkcollege)

♣ WEEK 37/Lecture 1 (September 9): An introduction: some of the keywords of this course; Theorem 1.50; the notion of topological chart and smooth compatibility of charts. Plan for the next time: the definition of manifolds (smooth atlases, smooth structure= maximal smooth atlases; first examples).

Blackboard photos of lecture 1.

The exercises for today's tutorial are:

read the lecture notes containing the reminders on Topology and Analysis and start having a first look at the definition of manifold.

♣ WEEK 37/Lecture 2 (September 11): Charts, smooth compatibility, smooth atlases, smooth structures, smooth manifolds, first remarks and examples (see the blackboard fotos). Plan for next time: reminder, finish the first remarks/examples then move on to other examples.

Blackboard photos of lecture 2.

The exercises for today's tutorial are:

1.24, 1.38, 1.53, Example 2.3, Example 2.4, 1.44, 1,48, 1.54.

Homework 1 (to be handed in by next Wednesday, by the start of the werkcollege):

♣ WEEK 38/Lecture 3 (September 16): Generous reminder on smooth structures/smooth manifolds, smooth maps, examples such as: (opens in) Euclidean spaces, embedded submanifolds of the Euclidean space and the RVT (regular value theorem), embedded submanifolds of general manifolds, then we focused on the sphere (including stereographic projections) and the projective space.

Blackboard photos of lecture 3.

The exercises for today's tutorial are:

2.37 (check that this implies 2.178), 38, 39, 49, 54/55 (point 1 and 3), extra: 2.52.

♣ WEEK 38/Lecture 4 (September 18): See the fotos below.

Blackboard photos of lecture 4.

The exercises for today's tutorial are:

Chapter 2: 185, 74, 26 or 28 + Continue working on exercises from previous weeks.

Homework 2 (to be handed in by next Wednesday, by the start of the werkcollege):

♣ WEEK 39/Lecture 5 (September 23): Tangent spaces.

Blackboard photos for this lecture.

The exercises for today's tutorial are:

2.31, 52, 2.213, 2.215, extra: 2.214.

Homeworks 3-4 (to be handed in by Monday, October 7th, by the start of the werkcollege).

♣ WEEK 39/Lecture 6 (September 25): Submersions, immersions, submanifolds.

Blackboard photos for this lecture.

The exercises for today's tutorial are: Fill in the details for the proof of Lemma 2.108 and do 2.110. Then the very important 2.114 and 2.121. Do also the following old homework from 2023 (but do it as an exercise, not as homework. Notice that some of the first questions are already covered by the previous list of exercises). More exercises may be give to you by the TAs; they will be announced on the "blackboard" page of the course.

Homework 3-4 (to be handed in by Monday, October 7th): see above.

♣ WEEK 40/Lecture 7(September 30): Tangent vectors (at p) as derivations (at p). Then the defintion of vector fields and the fact that they form a module over the ring/algebra os smooth (real-valued) functions.

Blackboard photos for this lecture.

The exercises for today's tutorial are: 2.189, 191, 198, 200 and 204 (want more practice? have a look at 2.111 or 188).

♣ WEEK 40/Lecture 8 (October 2): Lie derivatives $L_X$, vector fields as derivations, Lie bracket of vector fields, its properties and 1-2 computations, push forward of vector fields, brief intro on the Lie algebra of a Lie group (just "for fun"). And a brief advertisement of what will come next week: integral curves and flows of vector fields.

Blackboard photos for this lecture.

The exercises for today's tutorial are: 3.17 (hint 2.111), 3.22+3.23+3.24, 3.20, 3.25.

♣ WEEK 41/Lecture 9 (October 7): Integral curves; existence and uniqueness of maximal integral curves. Started (at the end) to talk about flows.

Blackboard photos for this lecture.

The exercises for today's tutorial are: 3.28, 3.32, 3.33, 3.31. See also this blackboard.

♣ WEEK 41/Lecture 10 (October 9; PLEASE NOTE THE CHANGE OF LOCATION: RUPPERT A):

Blackboard photos for this lecture.

The exercises for today's tutorial are: 3.64, then do item c of Exercise 3.103 and also compute all the integral curves of V, then 66, 67, 68.

Homework 5 .

♣ WEEK 42/Lecture 11 (October 14, BBG - 083): 1-forms

Blackboard photos BUT, this time, the writing on the blackboard was far from ideal ... certainly worse than in the previous year ... both the blackboard available was worse, as well as the teaching was worse due to some emergency which made me be late and rather disorganized .... and, for that reason, I am attaching here also the the blackboard photos from the previous year .

The exercises for today's tutorial are: 3.76,80,88,92 and 93.

♣ WEEK 42/Lecture 12 (October 16, PLEASE NOTE THE CHANGE OF LOCATION: BBG - 161):

Blackboard photos for this lecture.

The exercises for today's tutorial are:

Homework(to be handed in by next Wednesday, by the start of the werkcollege):

♣ WEEK 43/Lecture 13 (October 21, BBG - 079):

Blackboard photos for this lecture.

The exercises for today's tutorial are:

♣ WEEK 43/Lecture 14 (October 23, BBG - 205):

Blackboard photos for this lecture.

The exercises for today's tutorial are:

Homework(to be handed in by next Wednesday, by the start of the werkcollege):

♣ WEEK 44/Lecture 15 (October 28, BBG - 079):

Blackboard photos for this lecture.

The exercises for today's tutorial are:

♣ WEEK 44/Lecture 16 (October 30, we are planned for DDW - 1.30, we try to get a decent room instead):

Blackboard photos for this lecture.

The exercises for today's tutorial are:

Homework(to be handed in by next Wednesday, by the start of the werkcollege):

♣End exam: Tuesday, 5 November 2024, 17:00 - 20:00, EDUC - THEATRON.

IMPORTANT: PLEASE DO NOT RELY 100% ON THIS WEB-PAGE. I.E., IF YOU MISS A CLASS, PLEASE KEEP IN TOUCH WITH YOUR COLLEAGUES THAT WERE PRESENT, JUST TO MAKE SURE YOU STAY INFORMED.

Manifolds (Differentieerbare varieteiten) Blok 1, 2024/2025 (UNDER CONSTRUCTION)

THE SCHEDULE WEEK BY WEEK (I will try to add, after each lecture, a description of what was discussed in the lectures + the exercises from the lecture notes that you are supposed to do during the werkcollege)

IMPORTANT: PLEASE DO NOT RELY 100% ON THIS WEB-PAGE. I.E., IF YOU MISS A CLASS, PLEASE KEEP IN TOUCH WITH YOUR COLLEAGUES THAT WERE PRESENT, JUST TO MAKE SURE YOU STAY INFORMED.

Manifolds (Differentieerbare varieteiten)
Blok 1, 2024/2025 (UNDER CONSTRUCTION)