Representations of the Physical Universe

An upcoming book

Prerequisites

1) Old Notions Revisited

First draft completed, need to draw up figures.

The Cartesian Coordinate System

Linear Algebra and Coordinates

Euclidean vs. Affine Space

The naive inner product is not an invariant

Nonlinear Coordinates

Einstein's Summation Convention

Make clean and short

Exercises

2) New Horizons

Close to being done.

We want a section for defining Direct sums, Dual Spaces <-> functionals, and tensor product both INTUITIVELY and FORMALLY

The Manifold

Examples of Manifolds

Elementary Topology

Gaps in this section Add stereographic projection of sphere as example of coordinate charges

Embedded v.s Intrinsic

Vectors Reimagined

Multilinear Algebra: Views

What Follows

Exercises

Part 1: A Better Language

3) Differential Geometry

THINGS WE SHOULD HAVE DEFINED EARLIER ON:

C^\infty(M), That 1/k! on the k-form coefficient tensor

The Derivative and the Boundary

The 1-Form

The Exterior Algebra from the Wedge

Stokes' Theorem

Finish the proof here

Distance, a Metric

The Hodge Star and the Laplacian

Movement: Lie's Ideas

Finish some Lie derivative summaries

Exercises

A functional-programming type exercise to get a really clean compact way to define the exterior derivative

4) Harmonics: Fourier Analysis

Discrete, Bounded: Eigenvalues

Functions are very clearly n-component tuples forming a vector space

Continuous, Bounded: Fourier Series

Differentiation as a linear operator, finding its eigenvalues gives the fourier transform

Continuous, Unbounded: Fourier Transform

Harmonic Analysis on Higher Euclidean Space

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.

Harmonic Analysis on Graphs

Functions on graphs, graph laplacian. Motivate expanders Conclude with spectral graph theory

Exercises

What about real-valued eigenvalues giving rise to exponential decay eigenforms: c.f. the positive reals

5) Beyond Harmonics: Representation Theory

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

Part 2: Physics (In order of increasing difficulty)

6) SL(2,C), SU(2), SL(2,R)... the principles of finite dimensional QM

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

7) Classical Mechanics and Symplectic Geometry

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

8) Einstein's Gravity

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)

Part 3: More Advanced Topics (No preferred order)

8) Cohomology and Homology

Goal is a simple motivation of poincare duality and relating the space of harmonic p-forms to the (co)homology

9) An Introduction to Geometric Quantization

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. 

10) Classification of Simple Lie Algebras over C

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