GEOMETRY 2, FALL 2014

See also the course catalogue and (requires login) Absalon

SCHEDULE

Lectures:
Monday 10:15-12:00 Aud 8
Tuesday 13:15-15:00 Aud 5
Friday 9:15-10:00 Aud 8
Exercise classes:
Tuesday 15:15-17:00 room A104
Friday 10:15-12:00 room A101

LECTURE NOTES

Differentiable Manifolds, by Henrik Schlichtkrull. 2014 version.
ISBN 978-87-7078-458-0, Polyteknisk Boghandel.

LECTURES

The lectures are held by Henrik Schlichtkrull

(preliminary plan)
17/11 Monday 1.1-1.5
18/11 Tuesday 1.6-1.8
21/11 Friday 2.1-2.3
24/11 Monday 2.4-2.7
25/11 Tuesday 2.7-2.10
28/11 Friday 2.11
1/12 Monday 3.1-3.5
2/12 Tuesday 3.5-3.8
5/12 Friday 3.8-3.10
8/12 Monday 4.1-4.3
9/12 Tuesday 4.4-4.6
12/12 Friday 4.7-4.10
15/12 Monday 5.1-5.5
16/12 Tuesday 5.6-5.8
19/12 Friday 5.9-5.11
22/12 Monday No lecture. (See about a mandatory written assignment below)
2/1 Friday No lecture.
5/1 Monday 6.1-6.4
6/1 Tuesday 6.5-6.8
9/1 Friday 6.9-6.10
20/1 Tuesday 13:15-15:00 Aud 8 Question session.

EXERCISE CLASSES

Exercise classes are used for problem solving, and for presentation of exam questions by participants as preparation for the oral exam. There are 12 problem programs P1-P12 (see below), and 14 exam questions E1-E14 (also below). Further presentations will be planned once the exact number of students is known - the goal is that every student is offered the possibility to present.

The exercise classes are conducted by Massimilano Ungheretti

18/11 Tuesday No exercise class
21/11 Friday P1
25/11 Tuesday P2
28/11 Friday P3
2/12 Tuesday P4
5/12 Friday P5+E1
9/12 Tuesday P6+E2
12/12 Friday P7+E3
16/12 Tuesday P8+E4
19/12 Friday P9+E5+E6
2/1 Friday No exercise class. (See about a mandatory written assignment below)
6/1 Tuesday P10+E7+E8
9/1 Friday P11+E9+E10
13/1 Tuesday P12+E11+E12+E13+E14 Note: Double class: 13:15-17:00, room 1-0-37 (DIKU)
16/1 Friday. If there is interest: Selected exam question presentations. 10:15-12:00, room 1004 (DIKU)

MANDATORY WRITTEN ASSIGNMENT

A mandatory assignment replaces the teaching of December 22 + January 2. It will be web-posed on December 19, and must be returned by Tuesday January 6 at the start of the exercise class. An approved solution is necessary for participation in the exam.

PROBLEM PROGRAMS

Select problems from the following list. The problems cover the mentioned sections. Emphasis will be on the highlighted problems.
P1: 1.1-1.4, Exercises 1.1-1.7. 3,4,5,7
P2: 1.5-1.8, Exercises 1.8-1.17. 11,12,13,15,17
P3: 2.1-2.5, Exercises 2.1-2.10. 4,5,6,7,9,10
P4: 2.6-2.10, Exercises 2.11-2.16. All
P5: 3.1-3.6, Exercises 3.1-3.5. 2,3,4
P6: 3.7-3.10, Exercises 3.6-3.10. 6,7,8,10
P7: 4.1-4.4, Exercises 4.1-4.8. 2,3,4,5
P8: 4.5-4.8, Exercises 4.9-4.15. 10,11,12,13
P9: 5.1-5.6, Exercises 5.1-5.5. 2,3,4
P10: 5.7-5.10, Exercises 5.6-5.12. 8,9,10,11
P11: 6.1-6.5, Exercises 6.1-6.7. 1,3,4,6
P12: 6.6-6.9, Exercises 6.8-6.14. 10,12,13,14

EXAM

The oral exam is scheduled for January 22-23
The scope of the exam is as follows: The student draws a question from the following list, and is allowed approximately 30 minutes of preparation. The examination lasts 25 minutes, of which 15-20 minutes will be spent on the presentation. After that questions will be asked in the entire curiculum of the course. At the end the student receives a grade.
The official language of the exam is English (but Danish will be accepted as well).

EXAM QUESTIONS

E1: Manifolds in R^n
E2: Abstract manifolds. Projective space.
E3: The Grassmann manifold.
E4: The tangent space of an abstract manifold
E5: Smooth maps and their differentials
E6: Submanifolds
E7: Submersive maps and level sets
E8: The orthogonal group is a Lie group
E9: Partition of unity
E10: Embedding in Euclidean space
E11: The components of a manifold
E12: Vector fields and Lie brackets
E13: The Lie algebra of a Lie group
E14: The Lie algebra of GL(n,R)