5 credits, September-December 2006

Lecturer: Göran Bergqvist

Prerequisites:
MAI003 (NMAD11) Differential geometry. Basic knowledge of abstract
algebra, topology, and ordinary and partial differential equations

Literature: S. Morita,
Geometry of Differential Forms
(American Mathematical Society)

Supplementary reading: T. Frankel, The Geometry of Physics (Cambridge Univ. Press). I. Madsen and J. Tornehave, From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes (Cambridge Univ. Press). M. Nakahara, Geometry, Topology and Physics (Adam Hilger)

Organisation:
Lectures and student seminars

Examination:
Presentations and active participation

Schedule

13/9 15.15 (Determinanten) Short information meeting

20/9 10-12 (Kompakta Rummet) Lecture 1: Differential forms

27/9 10-12 (Glashuset) Lecture 2: Homology

28/9 10-12 (Glashuset) Student seminars. DY: The fundamental group and
coverings, IE&JJ: Lie groups and Lie algebras

6/10 10-12 (Glashuset) Student seminars. ZF&MH: Calculations of
homology groups, TB: Complex manifolds

11/10 10-12 (Glashuset) Lecture 3: De Rham cohomology

12/10 10-12 (Glashuset) Lecture 4: De Rham cohomology

19/10 10-12 (Glashuset) Lecture 5: Harmonic forms, Hodge decomposition

24/10 8-10 (Glashuset) Lecture 6: Hodge decomposition. Vector bundles

26/10 10-12 (Glashuset) Lecture 7: Vector bundles; connections and
curvature

31/10 10-12 (Glashuset) Student seminars. JJ: Invariant polynomials,
MH: Tangent bundles of projective spaces

2/11 10-12 (Glashuset) Lecture 8: Characteristic classes; Pontrjagin
and Chern classes

7/11 13-15 (Glashuset) Student seminars. TB&ZF: Euler classes

10/11 10-12 (Kompakta Rummet) Student seminars. IE&DY: Gauss-Bonnet
theorem

13/11 10-12 (Glashuset) Lecture 9: General fibre bundles, principal
bundles

16/11 10-12 (Glashuset) Lecture 10: Connections on fibre bundles

21/11 13-15 (Glashuset) Lecture 11: Curvature and characteristic classes

23/11 10-12 (Glashuset) Lecture 12: ... / Student seminars.
MH&DY: S^{1} bundles (sections and Euler
class)

28/11 10-12 (Glashuset) Student seminars. ZF: S^{1}
bundles (classification), JJ: S^{1} bundles
(curvature and Euler forms)

30/11 10-12 (Glashuset) Student seminars. TB: Chern-Simons forms, IE:
Stiefel-Whitney classes