Manifolds and Fibre Bundles
 

5 credits, September-December 2006

Lecturer: Göran Bergqvist

Contents: Homology and de Rham cohomology. Harmonic forms and Hodge decomposition. Lie groups and Lie algebras. Vector bundles, principal bundles and general fibre bundles; connections, curvature and characteristic classes.

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: S1 bundles (sections and Euler class)
28/11 10-12 (Glashuset) Student seminars. ZF: S1 bundles (classification), JJ: S1 bundles (curvature and Euler forms)
30/11 10-12 (Glashuset) Student seminars. TB: Chern-Simons forms, IE: Stiefel-Whitney classes


Göran Bergqvist