First draft completed, need to draw up figures.
The naive inner product is not an invariant
Make clean and short
Close to being done.
We want a section for defining Direct sums, Dual Spaces <-> functionals, and tensor product both INTUITIVELY and FORMALLY
Gaps in this section Add stereographic projection of sphere as example of coordinate charges
THINGS WE SHOULD HAVE DEFINED EARLIER ON:
C^\infty(M), That 1/k! on the k-form coefficient tensor
Finish the proof here
Finish some Lie derivative summaries
A functional-programming type exercise to get a really clean compact way to define the exterior derivative
Functions are very clearly n-component tuples forming a vector space
Differentiation as a linear operator, finding its eigenvalues gives the fourier transform
Here we derive the Laplacian, study diffusion, image processing, convolution Functions on 2D space as a limit of a 2D grid Laplacian is never obviously derived in any good text so derive it.
Functions on graphs, graph laplacian. Motivate expanders Conclude with spectral graph theory
What about real-valued eigenvalues giving rise to exponential decay eigenforms: c.f. the positive reals
We begin assuming knowledge of group theory/ intro-level abstract algebra. Begin with spectral graph theory motivating representations as generalizations of eigenvalues
Look at finite groups...
Build analogous results to fourier analysis
Return to graphs
graphs -> quivers?
Representation theory of k[x] = Jordan Normal Form
Define topological group.
Some statements about extending this to compact groups
Review how this ties together: rep theory of SO(3) leading to SO(3,1), and spinors (projective representations)
QI section
Tie this in to SU(n) but probably not past taht
What is the difference between velocity and momentum
Legendre transform
Feynman Argument for conservation in the Lagrangian viewpoint
Intuition behind how the symplectic form bundles position and momentum together
Lets us go from a Hamiltonian H -> \partial / \partial t -> the arrow of time -> A vector field
All we need is infinitesimal hamiltonian information, so gives an equivalence 1-forms <-> vector fields on T^*M
Covariant Derivatives. Motivate Riemann Curvature
READ WEYLS BOOK (or GR nut) it literally derives GR from symmetry
And then:
Black holes, let them spin let them be charged. Let there be parallel universe, everything.
Penrose diagrams
Expanding universes, redshift, Hubble's thing.. ADM might be too much its ugly imo (but the preceeding chapter HAS developed the hamiltonian formalism enough)
Goal is a simple motivation of poincare duality and relating the space of harmonic p-forms to the (co)homology
The idea should be that even though, physically, we can barely grasp quantization at first glance, the mathematics guides us and we shuold try to interpret THAT physically.
You Qi's notes, together with Humphreys will be the guide
The goal is to introduce, with reasonable proof, the ABCDEFG classification of Lie Algebras