"Einstein himself regarded the abstract four-manifold as what remains of the 'ether' in general relativity... Perhaps we may say that in studying smooth manifolds we are studying the possible shapes of the ether." - R.W. Sharpe
Differential topology uses differential and integral calculus to study curves, surfaces, and their higher dimensional counterparts, collectively called smooth manifolds.
This is an advertisement for a guided reading seminar for the Spring 2022 semester on differential topology, with readings taken from:
Milnor, Topology from the Differentiable Viewpoint,
Guillemin and Pollack, Differential Topology,
Spivak, Calculus on Manifolds,
Penrose, The Road to Reality, and
Ueno, Shiga, and Morita, A Mathematical Gift.
We meet in Shineman 176, 4-5pm, on Fridays. All are welcome to attend.
A link to the seminar texts is here.
Begin reading draft of "Differential forms and the de Rham algebra." (Sent to you in an email.)
Continue readings from last week.
Begin working through the notes "Graded algebra": contravariant and covariant tensors; alternating tensors; tensor and exterior algebras.
Review of dual spaces, independent functions, and manifolds "cut out" by functions; transversality.
Manifolds in physics; the non-euclidean nature of configuration spaces; "What is topology?"; real projective spaces; loops, lifts, and covering spaces; fundamental groups.
Smoothly varying families of manifolds; smooth deformations; preimages and fibers; Local Submersion Theorem; Preimage Theorem; Lie groups and Lie algebras; manifolds "cut out" by functions.
The very last bit on derivatives of smooth maps; the Local Immersion Theorem in earnest; images of immersions; proper maps and embeddings.
More on derivatives as linear approximations; more on derivatives of smooth maps of manifolds; the beginnings of the Inverse Function Theorem and Local Immersion Theorem.
Derivatives as directional derivatives and linear approximations; Jacobian matrices; tangent spaces; derivatives of smooth maps of manifolds; the chain rule.
More on the definition of a manifold; some big-picture questions; concrete examples.
Primer on set theory and functions; definition of a manifold.