Mathematics / research
Bundles, connections, and characteristic classes
Study global twisting and gauge geometry through principal bundles, curvature, obstruction classes, and Chern-Weil theory.
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Inside this module
Module 01 research
Bundles, connections, and characteristic classes
Study global twisting and gauge geometry through principal bundles, curvature, obstruction classes, and Chern-Weil theory.
- 01
Principal bundles and transition functions
Build a global torsor from local sections while keeping the order of every transition map explicit.
24 min - 02
Associated vector bundles and frame bundles
Translate between a torsor of frames, representation-valued coordinates, and intrinsic vector-bundle sections.
25 min - 03
Connections as horizontal distributions
Derive connection forms, their affine overlap law, and the parallel-transport equation from a horizontal splitting.
27 min - 04
Curvature, the Bianchi identity, and holonomy
Compute the obstruction to commuting covariant derivatives and separate infinitesimal curvature from global loop transport.
28 min - 05
Gauge transformations and connection moduli
Separate frame changes from geometry and compute a moduli space where large gauges matter.
26 min - 06
Classifying spaces and universal bundles
Turn bundle isomorphism questions into homotopy questions while retaining the hypotheses that make classification valid.
28 min - 07
Euler and Stiefel–Whitney classes
Use integral and mod-two obstructions without confusing orientation, stable triviality, and actual triviality.
28 min - 08
Chern and Pontryagin classes
Compute integral characteristic classes through formal roots and carefully distinguish complexification from underlying real bundles.
29 min - 09
The Chern–Weil homomorphism and invariant polynomials
Build closed global forms from curvature, with explicit normalizations and an honest account of lost torsion.
30 min - 10
Chern–Simons forms and transgression
Derive the exact difference between two characteristic forms without mistaking a local primitive for a global trivialization.
31 min