Gauge theory and topology: an introduction (Pisa, Fall 2025).

Syllabus and a monopole.

Part I: geometry, topology and analysis in gauge theory.

• 1 - From magnetostatics to gauge theory.
• 2 - Connections, curvature and topology.
• 3 - Yang-Mills theory on Riemannian manifolds.
• 4 - Instantons, monopoles and vortices.
• 5 - Elliptic operators and Sobolev spaces.
• 6 - The Hodge theorem and index theory.

Part II: the vortex equations on a surface.

• 7 - The Yang-Mills-Higgs functional on a surface.
• 8 - Geometry of holomorphic line bundles.
• 9 - Moment maps in symplectic geometry.
• 10 - Moduli spaces in gauge theory.
• 11 - Vortices and symmetric products.
• 12 - Seiberg-Witten theory and Floer homology.

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