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This is an attempt at a bottom-up approach to modern differential geometry, covering deep topics in maximum detail with minimal prerequisites. Too often in math education, fundamental concepts are treated as a mere "means to an end" (such as proving an important theorem), so the original motivation behind the introduction of these concepts gets lost, making them very unintuitive and dry. Therefore, "autistic" in the subtitle refers to the principle: no sophisticated construction is introduced until it is motivated enough to feel necessary.
Prerequisites: although a short reference section on point set topology is included, it is not comprehensive. Thus, apart from general calculus, some familiarity with basic topology definitely helps. In certain applications and examples, other subjects such as PDEs, complex analysis, or physics will come in handy.
Reading guidance: top-level ``parts'' of the book (Roman numerals) are well separated in themes so that cross-references between them are pretty minimal. Thus, for example, if you are broadly familiar with the topics in parts 1-3, you are probably safe reading from part 4.
If you would like to support the continuing work on this book, consider buying me a coffee: ☕ https://buymeacoffee.com/abogatskiy ☕
Topics covered (the ones with ellipses are under construction)
- Category theory
- Homotopy theory
- Smooth manifolds
- Covering spaces
- Lie groups and algebras
- Fiber bundles
- Lie group actions
- Cartan's approach to PDEs (Exterior Differential Systems)...
- Curves and Surfaces
- Principal connections
- Cartan connections...
- Riemannian geometry, Morse theory...
- Complex geometry...
- Symplectic geometry...
- Homological algebra
- Homology and cohomology of topological spaces
- Homology and cohomology of smooth manifolds (de Rham, Hodge, Morse)...
- Sheaf cohomology...
- Structure theory of Lie groups and algebras...
- Classification of fiber bundles, characteristic classes...
- Gauge theory, Dirac bundles...
- Index theorems...