Manifolds (Differentieerbare varieteiten)

Blok 1, 2021/2022

This is the web-site for the course "Differentieerbare varieteiten" given in blok 1, Fall 2021. Here you will find all the practical informations about the course, changes that take place during the year, lecture notes, etc. I will try to make use also of the blackboard.
THE LECTURES:

EXCEPTIONS:

  • for lecture 4 (September 27) and lecture 5 (September 29), we will switch with the werkcolleges, i.e. the lectures will be at 9:00 AM (same location).
  • for lecture 10 (October 13) the werkcollege will be in KBG ATLAS, while the werkcollege in MIN 2.0.

    LECTURER: Marius Crainic.

    THE WERKCOLLEGES:

    TEACHING ASSISTANTS: Anna Fokma, Lucas Smits, Mick Schilder.

    HOMEWORKS: Each Monday, in the afternoon, you will receive a homework that you have to "hand in" by the next Monday, 9AM. If you are not present at the wekcollege on Monday please discuss with the TAs in advance about the way you can hand in your homework. Please do not send your homework to the lecturer (then your homework may get lost ...).

    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:

    Lecture notes: Please be aware that the lecture notes may still contain typos. So, if there is something that you do not understand, please ask it at the werkcollege; also, if you find typos, please communicate them to the teaching assistants. This will be of great help to improve the lecture notes and make them into a regular "dictaat" that can be printed for students.

    I will be using this set of lecture notes (they are the notes from last year to which I added, at the end, the exams from the last year).

    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 13): Reminder on Topology and Analysis.

    Werkcollege (September 13 and September 15): read the lecture notes containing the reminders on Topology and Analysis.

    Homework (for the next week on Monday): see here.

    ♣ WEEK 37/Lecture 2 (September 15): Continuation with the reminder. Then the notion of change of coordinates between two charts, the (smooth) compatibility of two charts, the notion of smooth atlas and smoothness of real-valued functions w.r.t. a given atlas (i.e. section 2.1.1).

    Suggested exercises for the next werkcollege: 1.52, 1.54, 1.58, 2.5.

    ♣ WEEK 38/Lecture 3 (September 20): Smooth structures, the smooth structure induced by an atlas (not necessarily maximal), smooth manifolds, smooth maps (between smooth manifolds), diffeomorphisms. The examples mentioned so far: $\mathbb{R}^m$ with various atlases, $S^2$ with various different atlases (given by orthogonal projections and/or stereographic projections), embedded submanifolds of $\mathbb{R}^m$ in the sense of Analysis, opens inside manifolds, product of manifolds.

    Suggested exercises for the next werkcollege: 2.9, 2.21, 2.31, 2.38, 2.46.

    Homework: 2.60.

    ♣WEEK 38/Lecture 4 (September 22): Reminder, submersions and immersions (again) and then mainly examples, most notably the projective spaces and maps between them,

    Suggested exercises for the next werkcollege: 2.52, 2.84, 2.6, 2.86, 2.65 and if you want to have more fun with the Hopf fibration look also at 2.70, 2.87 parts a, c and d.

    Reminder: next week (Sept 27 and 29) the lectures will be at 9 AM!!!

    ♣ WEEK 39/Lecture 5 (September 27 at 9:00!): and the example with O(n), general embedded submanifolds, embeddings.

    Suggested exercises for the next werkcollege: First the left over from the last week (make sure you did at leasr 2.86 or 2.87) and then 2.75, part 1 of 2.76, then 2.82, 2.89.

    Homework, to be handed it on Wednesday, October 6th (since it was given only on Tuesday afternoon): here.

    ♣WEEK 39/Lecture 6. (September 29 at 9:00)): Other types of submanifolds (immersed, with unique smooth structure, initial). Then a bit about Lie groups.

    Suggested exercises for the next werkcollege: make sure you did 2.89 and 2.95. Then 2.110. Then recap the tangent space for submanifolds of Euclidean spaces $R^n$ (page 25).

    ♣ WEEK 40/Lecture 7 (October 4): The (abstract) tangent spaces, via charts, and also the definition and the first remark of the approach via derivations.

    Exercises for the next werkcollege: must do: 3.6, 3.7, 3.12. Then 3.10 and as much as you can do from 3.14, in the following order: 1, 3, 2.

    Homework: here.

    Bonus exercise: Exercise 2.87. Note: the bonus exercise, if solved completely, will add 0.5 points to the final mark. However, we accept the bonus exercise only if you handed in also the homework of the week.

    ♣WEEK 40/Lecture 8 (October 6): Finished the discussion on tangent spaces: i.e. discussed in more detail also the approach via derivations and described the way to go from one description to the other, in a 1-1 way.

    Suggested exercises for the next werkcollege: before the pause look at 3.24 and start reading the conclusions on tangent spaces as summarized in subsection 3.4. In particular the last version of the regular value theorem (Theorem 3.26). What is new in this version is the very last part, about the tangent spaces. Try to find a simple proof of this last part, without using charts. E.g. prove first the direct inclusion $\subset$ using that tangent vectors can be realized as speeds of curves (we gave such an argument last week, for $N= \mathbb{R}^n$). To finish the prove, to a dimension counting. Finally, to see how to use this theorem, please do (a), (b) and (c) of exercise 2.87.

    ♣ WEEK 41/Lecture 9 (October 11): Vector fields.

    Suggested exercises for the next werkcollege: 3.38, 3.43, 3.44, 3.40 and 3.48?

    ♣WEEK 41/Lecture 10 (October 13 in Min 2.0!!!): (Maximal) integral curves.

    Suggested exercises for the next werkcollege: 3.75, 3.79, 3.78, and you can also discuss some of the statements made on the sheet with the homework (see the next link), which are present whenever you see the colored werkcollege.

    Homework: here. .

    ♣ WEEK 42/Lecture 11 (October 18): We finished vector fields, flows, Lie derivatives.

    Suggested exercises for the next werkcollege:

    ♣WEEK 42/Lecture 12 (October 20): 1-forms.

    Suggested exercises for the next werkcollege:

    Homework: here.

    ♣ WEEK 43/Lecture 13 (October 25): Differential forms of arbitrary degree.

    Suggested exercises for the next werkcollege: 4.4 and 4.5, 4.37, and then 4.39 and 4.38 in this order.

    ♣WEEK 43/Lecture 14 (October 27):

    Suggested exercises for the next werkcollege:

    Homework: TBA.

    ♣ WEEK 44/Lecture 15 (November 1): Volume forms, orientations (defined "directly" using volume forms, without the extra-discussion from the notes), integration.

    Suggested exercises for the next werkcollege:

    ♣WEEK 44/Lecture 16 (November 3): A bit of DeRham cohomology.

    Some pictures with examples on the blackboard (during the last werkcollege) on how to check that a form is a volume form, and how to check whether a chart is positively oriented or not: picture 1 , picture 2 , picture 3 , picture 4 .

    Homework: TBA.

    ♣End exam: November 8, 11:30, BBG 109/EDUC THEATRON.

    IMPORTANT: PLEASE DO NOT RELY 100% ON THIS WEB-PAGE. MORE PRECISELY, ALTHOUGH I DO TRY TO KEEP IT UPDATED (E.G. BY SPECIFYING WHICH IS THE HOMEWORK), SOMETIMES I AM NOT ABLE TO DO IT. IN SUCH CASES, PLEASE CONTACT YOUR COLLEAGUES THAT WERE PRESENT AT THE LECTURE OR WERKCOLLEGE.