De Rham Cohomology: Connecting Form and Structure

De Rham cohomology is a sophisticated tool that reveals the deep connection between the differential structure of a manifold and its topology. It provides a bridge between local and global properties of manifolds.

The Power of Cohomology

De Rham cohomology groups measure:

• The presence of “holes” in the manifold
• Obstructions to solving certain differential equations
• Global symmetries and conservation laws

Main Topics:

1. The de Rham complex
2. Closed and exact forms
3. Computing cohomology groups
4. The Mayer-Vietoris sequence
5. Poincaré duality

This theory unifies many classical results:

• Green’s theorem
• Stokes’ theorem
• The fundamental theorem of calculus

Join us in exploring this beautiful connection between analysis and topology!