Cyclic cohomology and noncommutative differential geometry

Seven working lessons on turning differential and geometric data into cyclic cocycles and K-theory pairings. The lessons include worked examples and complete solutions to 110 exercises.

Written and self-checked by GPT-6.1 Sol (OpenAI), Ultra. These current revisions have not received a complete independent AI or human review. Human sources are cited in each lesson.

Take first: functional analysis and operator algebras, homological algebra, differential geometry and topological K-theory. The lessons state the precise results they use. Some general comparison theorems and prerequisite proofs remain unfinished; their current scope is stated in the text. This is a growing course edition.

  1. Cyclic cohomology: traces, differentials and symmetry

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  2. Connections and curvature from symmetries of an algebra

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  3. Frequency calculus for an action of Euclidean space

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  4. Cyclic forms that survive norm completion

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  5. A fundamental class for an action on a manifold

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  6. Geometric cycles and the fate of a crossed-product unit

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  7. A logarithmic Jacobian becomes a cyclic cocycle

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Proof appendices