MA3407 Introduction to Lie theory
Special question and Exercise session Monday 02.12. 10-12 SBII 656
There is no set program for the question and exercise session. So come with your own questions. If nobody turns up it will not take place.
| Schedule | Room | |
|---|---|---|
| Lecture | Mon 12.15-13.45 | GL-RFB D4-132 |
| Tue 12.15-13.45 | GL-RFB R92 | |
| Exercise session | Mon 8.30-10.00 | moved to office 1338 |
| Alexander Schmeding | Office 1338 SBII | |
| Email: alexander [dot] schmeding [at] ntnu [dot] no | ||
Exam:
Oral examinations scheduled for the first two weeks of december.
All oral examinations take place in the Office 1338 SBII.
Course content - What is Lie theory?
In the first year mathematics courses, students meet analysis (including simple differential equations) and (linear) algebra (for example in the guise of vector spaces or as groups encoding symmetry). The topic of the course MA3407, Lie theory, sits at a meeting point for all of these ideas.
Lie theory deals with the structure and properties of Lie groups and Lie algebras. These are named after the Norwegian mathematician Marius Sophus Lie (1842 in Nordfordeit, Norway - 1899). Sophus Lie investigated groups of symmetries associated to differential equations and called these groups "continuous groups" (a term which nowadays has been replaced with the term "Lie group"). It turns out that by blending groups with the requirement that the group operations are differentiable in a suitable sense, yields a rich class of objects, the so called Lie groups. These groups are intimately connected to a on first glance much simpler structure, the Lie algebra. Lie theory investigates the interplay between the non-linear structure of a Lie group and the linear structure of a Lie algebra. The course will give an introduction to this theory and its applications.
In the first part of this course we will investigate the important special case of matrix Lie groups (= certain groups of matrices). These arise in many applications from numerical analysis to mathematical physics, mechanics and geometry. Matrix Lie groups have the advantage that all elements of Lie theory are easily accessible in these examples without the need of more advanced tools. In the second part of the course we will then develop the ideas from Lie theory and investigate the relation of Lie groups with geometric structures. Time permitting this will lead to the beautiful theory of Lie group actions on manifolds and Lie groups as symmetry groups of geometric structures.
See also the study handbook for more information on course content.
Course material
Lecture notes (will be continuously updated)
Course logbook
| Date | What | Recommended Exercises |
|---|---|---|
| 20.08 | Finished Introduction and Section 1.1 | Exercises at end of Section 1.1. |
| Slides from the lecture. | Check Appendix A.1 and A.2 we will not go through them in class. Basic exercises from A.1 are recommended | |
| 26.08 | Section 1.2, Lemma B.0.1 | Exercises at the end of section 1.2 and 1.3 |
| 27.08 | Section 1.3 | slides for week 2 |
| 02.09 | Section 1.4 (stopped at 1.4.9) | Exercises in section 1.4, In case you are really ambitious read up on the ODE solution theory we use |
| 03.09 | Appendix A.3 | slides for week 3 |
| 09.09 | Rest of section 1.4, parts of Appendix A.2 | Exercises in section 1.5 |
| 10.09 | Section 1.5, Section 1.6, up to and including 1.6.2. | Slides for week 4 |
| 16.09 | Section 1.6 and 1.7 up to and including 1.7.2 | Exercises in section 1.6 and 1.7, Exercise B.1.1-B.1.2 |
| 17.09 | Section 1.7 | Slides for week 5 |
| 23.09 | Section 1.8 | Exercises in section 1.8 and Exercise 1.9.1 |
| 24.09 | Section 1.9 | Slides for week 6 |
| 30,09 | Section 2 up to and including 2.1.8 | Exercises in Section 2.1 |
| 01.10 | Section 2.1 | Slides for week 7 |
| 07.10 | Section 2.2 up to 2.2.6 | Exercises in Section 2.2. |
| 08.10 | Rest of section 2.2 | Slides for week 8 |
| 14.10 | Section 2.3 up to 2.3.9 | Exercises in Section 2.3 |
| 15.10 | Section 2.3 and 3 up to 3.0.3 | Slides for week 9 |
| 21.10 | Section 3.0 | Exercises in 3.0 |
| 22.10 | Appendix C, Section 3.1 up to 3.1.4 | Slides for week 10 |
| 28.10 | Section 3.1 and Section 3.2 up to 3.2.2 | Exercises in Sections 3.1 and 3.2 |
| 29.10 | Section 3.2 and Lemma 3.3.1 | Slides for week 11 |
| 04.11 | Section 3.3. up to and including 3.3.10 | Exercises in Sections 3.3 and 3.4 |
| 05.11 | Section 3.3 and 3.4 and 4.0.1 | Slides for week 12 , V. Arnold on teaching mathematics |
| 11.11 | Section 4.0 | Exercise in Sections 4.0 and 4.1 |
| 12.11 | Section 4.1 | Slides for week 13 |
| 18.11 | Video lecture! Start with 4.2 Overview in the Youtube Playlist | Exercises in Section 4.2 (the Exercise concerning Whiteheads Lemma is not recommended but left for ambitious students) |
| 19.11 | Section 4.2 |
References
The course will loosely follow parts of
Hilgert, J. and Neeb K.-H.: Structure and Geometry of Lie groups, Springer 2012 (Download via the library and NTNUs Springer Link subscription for free)