Multivectors, spinors and index theorems

Lecturer: Andreas Rosén, andreas.rosen@chalmers.se

Course description

Link to course plan

Notices:

20/11:
The course is finished. Please note that if you have feedback of any kind on the book, I VERY much appreciate if you share it with me! This will improve the next version.

17/11:
Last set of home work problems posted below. Deadline for that: 28/11.

11/11:
Three lectures left. We plan to finish the course next week. Information about the oral exam is posted below.

6/11:
Home work problems 6 posted below.

27/10:
Home work 5 posted below, with deadline next week.

2/10:
Home work 1 corrected. I hand them back on Monday. See below for information about grading, and hints for the home works problem.

23/9:
Three lectures this week, all in MVH11 (see below). No lectures next week.
Home work problems, set 2 and 3, are posted below.

15/9:
No lecture today, due to information meeting for PhD students.

5/9:
You find the first set of home work problems below.

1/9:
Course literature "Geometric multivector analysis" is now available at DC= DistributionsCentralen. (Somewhere a little south from Hörsalarna, and west from Gibraltargatan.)

21/8:
Welcome to the web page for this PhD course. First thing is to decide when to schedule the lectures: This is to be decided on our first meeting 1/9 at 13.15 in room Pascal: Please bring your calendar.

Lectures:

The following plan for the lectures will be continuously updated during the course.

Dates	Contents of lectures	Sections in the book	Deadlines and other to-do:s
Lecture 0: 1/9 room Pascal	Info about the course and examination An informal overview of the course: What are multivectors, spinors and Dirac operators? What are the index theorems that we aim to prove?	1.1-1.3	Get a copy of the course book: Available from DC (=Distributions- Centralen) at course start.
Lecture 1: 4/9 10.00-11.45 MVH11	Construction of multivectors, universal property Grassmann cone, oriented measure	2.1-2.4 1.5
Lecture 2: 8/9 13.15-15.00 MVH11	Duality, inner product spaces Multicovectors and alternating forms Interior products and Hodge stars	2.5-2.8
Lecture 3: 11/9 10.00-11.45 MVH11	The Clifford product Three dimensional rotations, quaternions and spin	3.1-3.2
Lecture 4: 18/9 10.00-11.45 MVH11	The Clifford cone Euclidean rotations and spin groups Infinitesimal rotations and bivectors	4.1-4.3	Home work 1: exterior algebra
Lecture 5: 22/9 13.15-15.00 MVH11	Abstract Clifford algebras Representation of real Clifford algebras	3.3-3.4
Lecture 6: 23/9 10.00-11.45 MVH11	Complex spinors in inner product spaces The standard representation Ideals in matrix algebras	5.1-5.2 1.4, 1.6
Lecture 7: 24/9 10.00-11.45 MVH11	Uniqueness up to almost unique isomorphism Mappings of spinors	5.2-5.3
Lecture 8: 6/10 13.15-15.00 MVH11	Exterior derivative and pullback Interior derivative and pushforwards	7.1-7.2	Home work 2: Clifford algebra
Lecture 9: 9/10 10.00-11.45 MVH11	Integration of k-covector fields and general k-forms Embedded oriented k-surfaces Stokes theorem	7.4-7.5 6.1	Home work 3: spinors
Lecture 10:13/10 13.15-15.00 MVH11	Elliptic Dirac operators and hypercomplex analysis Maxwell's equations	9.1-9.2
Lecture 11: 16/10 10.00-11.45 MVH11	The tangent bundle Levi-Civita covariant derivative Frames Covariant derivative 1-forms The exterior bundle, its covariant derivative	6.2-6.4 7.3 11.1
Lecture 12: 20/10 13.15-15.00 MVH11	Hyperbolic Dirac operators Maxwell's equations, Dirac's equation	10.1-10.2
Lecture 13: 24/10 13.15-15.00 NOTE: Friday! MVH11	Exterior and interior derivatives on manifolds Curvature 2-forms Commutator formulas for covariant derivative and curvature	11.1-11.3	Home work 4: affine multivector calculus
Lecture 14: 27/10 13.15-15.00 MVH11	Hodge decompositions Gaffney inequality Weitzenböck, Laplace-Beltrami operator	11.4-11.5 8.2
Lecture 15: 30/10 10.00-11.45 MVH11	Cech Cohomology Algebraic calculation of Betti numbers Poincaré's theorem	11.6 8.8-8.9 8.1
Lecture 16: 3/11 13.15-15.00 MVL14	Chern-Gauss-Bonnet 1 The heat equation method Normal coordinates	12.1-12.2	Home work 5: multivectors and manifolds
Lecture 17: 10/11 13.15-15.00 MVL14	Chern-Gauss-Bonnet 2 Trace calculations Pfaffian	12.2
Lecture 18: 13/11 10.00-11.45 MVL14	Spinor bundle Covariant derivative and curvature Stiefel-Whitney classes	12.3	Home work 6: Chern-Gauss-Bonnet
Lecture 19: 17/11 13.15-15.00 MVL14	Atiyah-Singer 1 spin-Dirac operator Weitzenböck formula	12.3-12.4
Lecture 20: 20/11 10.00-11.45 MVL14	Atiyah-Singer 2 Mehler's formula The A-roof functional	12.4	Home work 7: Atiyah-Singer

Examination:

For PhD students taking the course, the examination consists of

Home work assignments. There will be 7 sets of home work problems, with deadlines as above. I grade each solution as follows: 0=not pass, 1= pass, not without problem, 2= pass, 3= excellent solution. At the end of the course, and before the oral exam, you will need to hand in a new solution to each problem judged 0=not pass.
An oral exam. Before this you need to have passed the home work assignments. Here are details of the oral exam containing a detailed list of requirements.

Home work problems. This file is updated during the course. Last update: 4/12.

About one week before each deadline, the version will be final for that set of home works.
Home work 1: above version is final.
Hints:
1: Exc. 2.30, 2.46, 2.47, 7.54
2: Cor. 2.69, Exc. 2.73
3: Prop. 1.40
Home work 2: above version is final.
Hints:
1: Prop. 3.17: The key problem: what multiple of pi is b unique modulo?
2: Lem. 4.22, Ex 4.24 and 4.25
4: Exc. 3.31, Lem. 3.45 + recall from lecture 5 the representation of the euclidean line.
Home work 3: above version is final.
Hints:
1: Thm. 5.5, Props. 5.16-18
2: Prop. 5.24, comment after Prop. 5.15
Home work 4: above version is final.
Hints:
1: Why is the formula valid for general MULTVECTOR fields?
3: Prop. 7.8(i)
Home work 5: above version is final.
Hints:
1. The dual basis may be useful.
2: Which of the two dualities requires orientablility for its global formulation?
3. For a bounded and bijective linear operator T:X->Y between Banach spaces: What does functional analysis say about its inverse?
Apply this result for suitable T, X and Y. Be careful to say which norms you use! When is Hodge used?
Home work 6: above version is final.
1: Prop. 1.43 can be used for a short proof.
2. Bianchi symmetries? Prop. 3.12? ???
3: Note: any metric on the given manifold will give the same CGB integral, right?
Home work 7: above version is final.
1: Note: it is meaningless to say that an n-vector field on a manifold is constant. But you want to show that such a field has zero covariant derivatives.
3: If the y-integral of k(x,y) is 1 with almost all the mass in a small neighbourhood of x, then int k(x,y)f(y)dy is roughly...