This is a graduate-level introduction to differential manifolds (mainly on smooth manifolds).
Teaching assistants: 何玟辛，李进钊.
Final Score = 30% Homework + 30% Midterm Exam + 40% Final Exam.
September 23, 2024,
Topics: Introduction to this course, definition of manifold, Basic examples of manifolds.
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September 24, 2024,
Topics: Construct manifolds: product, open subset, quotient, basic examples of Lie groups.
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September 29, 2024,
Topics: Group action (by Lie groups), reduction (example: Grassmannian), definition of vector bundle (example: Möbius bundle).
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September 30, 2024,
Topics: Construct vector bundles (example: tangent bundle, cotangent bundle), sections (example: vector fields), Poincaré-Hopf theorem, directional derivative (of a function on a manifold), definition of connection.
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October 8, 2024,
Topics: Connection, Bracket (of vector fields), tensor (of vector spaces), universal property.
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October 14, 2024,
Topics: Tensor algebra, tensor bundle, tensor field (example: metric tensors), non-example of tensor fields (connection, bracket), Sym vs. Alt, wedge algebra (definition + basis).
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October 15, 2024,
Topics: Wedge product, Hodge star operation, wedge bundle, k-form (as sections of the wedge-k bundle), exterior derivative (formula), dd = 0, interior multiplication.
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October 21, 2024,
Topics: Examples in calculus of dd =0 (grad, curl, div), Maxwell's equations (via forms), definition of flowlines (examples: 2-torus sitting in \R^3, constructing symplectic matrices).
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October 22, 2024,
Topics: Completeness of vector fields, time-dependent vector fields, 1-parameter family of diffeomorphisms, equivalence to vector fields property, 2-parameter family of diffeomorphisms (example: related to bracket).
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October 28, 2024,
Topics: Pushforward and pullback (of diffeomorphisms), Lie derivative (with input vector fields and forms), computational examples of Lie derivative, Cartan's magic formula.
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October 29, 2024,
Topics: Definition of a submanifold (example: graph of a smooth map), submanifold is a smooth manifold itself, closed-subgroup Theorem, pushforward of a smooth map (between possibly different manifolds).
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November 4 and November 5, 2024 (lectured by Professor Yu LI),
Topics: Rank of a smooth map, critical point & value of a smooth map, regular point & value of a smooth map (example - higher genus closed surface), immersion and submersion (definition, example), Constant Rank Theorem (and many of its corollaries), embedding (definition), embedding of F = embedded submanifold, various embedding theorems.
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November 11, 2024,
Topics: Isometric embeddings, covariant/contravariant functors (example: pushforward and pullback), (vector) bundle map, short exact sequence of vector bundles (example: normal bundle and conormal bundle), transversality of submanifolds, transversality of maps (example: fiber bundle).
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November 12, 2024,
Topics: Fiber bundle (universal property), "bump" function (construction), partition of unity (P.O.U.), three applications: bump function over any closed subset, smooth extension, "compact" version of Whitney embedding theorem.
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November 18, 2024,
Topics: Definition of integration, (no boundary version of) Stokes' theorem, manifold with boundary, orientability of the boundary, examples of manifold with boundary (sublevel set), Brouwer's fixed point theorem (application: eigenvalue of positive-entry matrix).
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November 19, 2024,
Topics: Orientation, induced orientation on the boundary, computation of integration via parametrization, Stokes' Theorem (example: recover the Green formula and the Gauss formula in calculus), definition of divergence (on a manifold), Divergence theorem (as a generalization of the Gauss Theorem).
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