hey nerdinthearena,

i too find this area to be fascinating and wish i knew more on the upper end myself. i'm just going to list off a few resources. in my opinion, graduate school will concentrate a lot on progressing your technical knowledge, but will likely not give you a lot of time to hone your intuition (at least in the first few years). so, the more time you spend in undergraduate school doing so, the better.

helpful for intuition and basic understanding

• geometrical vectors by gabriel weinreich - why there's more to vectors and how to draw them. this basically shows visual examples of differential forms and the like.
  
• advanced calculus: a differential forms approach by harold edwards - a fantastic approach to differential forms. it's a treasure, so just read it. :) it is actually well built for skimming and skipping around.
  
• vector calculus, linear algebra, and differential forms: a unified approach - this is a big book, but you could just jump ahead to the later sections covering differential forms for an interesting approach and technical skill building.
  
• div, grad, curl are dead by william burke - pure intuition and similar to geometrical vectors. don't look for anything else here, but definitely read it.
  
• applied differential geometry by william burke - a more complete work than the above and showcases a lot of the connections between geometry and physics.
  
  

  more advanced but still intuitive
  

• an introduction to manifolds - this is at the same level and covers essentially the same material as lee, but in my opinion, it is much better. it is much more concise and to the point with more focused exercises.
  
• differential forms in algebraic topology by raoul bott and loring tu - the sequel to the above, but i haven't read beyond the first chapter. it is well known as a must read classic, but having some topology background will help with this book.
  
• the laplacian on a riemannian manifold: an introduction to analysis on manifolds - an interesting introduction to geometric analysis.
  
• differentiable manifolds: forms, currents, harmonic forms by georges de rham - a classic by one of the greats and also one of the only sources i've found that covers de rham currents (besides research papers that just use them).
  
• riemannian manifolds: an introduction to curvature by john lee - the sequel to lee's smooth manifolds book.
  
  hopefully this helps. if i were to revisit geometric analysis, i would basically use the above books to help bone up my understanding, intuition, and technical skill before moving on. these are also mainly geometry books, so learning analysis (like functional analysis) would be good as well. i mainly have three suggestions there.
  
  

  three general analysis favorites
  

• introductory functional analysis with applications by erwin kreyszig - used everywhere.
  
• calculus of variations by gelfand and fomin
  
• linear differential operators by cornelius lanczos - a geometrical look at the subject. also take a look at the variational principles of mechanics by him as well.

That's pretty cool. Unfortunately finding geodesics is a pain because you end up trying to solve evil nonlinear systems of differential equations. This is a great book if you're interested in learning some more about calculus of variations. If you have any questions I can try to answer them.

For Variational Calculus, the best references are Landau and Lifchitz and Gelfand and Fomin. The former is really a mechanics book that incorporates variational calculus in a very rigorous manner that one would expect from a theoretical physicist. The latter is a straight-up variational calculus book. Both are relatively cheap (you can find landau for cheaper than the amazon price).

For non-commutative geometry, there is this classic paper. /u/hopffiber gave the classic references for the rest of the topics, although you should think about learning quantum field theory since all the applications of Lie algebras come from QFT and String Theory. There are some excellent notes by David Tong that you can find with google-fu.

Have you had a rigorous course on Analytical Mechanics? You will learn all about Noether's theorem there. How does Noether's theorem relate to charge conservation? For ANY continuous symmetry of the Lagrangian, we observe an associate Noether current made up of Noether charge. A continuous symmetry of the lagrangian is a symmetry that is generated by that lagrangian's Lie group. For example, the Lie group associated with electricity and magnetism is U(1). U(1) is the unitary group in 1 dimension and represents complex rotations in 1D. This is equivalent to SO(2), the 2 dimensional rotations in real space. If you apply this symmetry to the electromagnetic lagrangian using the proper covariant derivatives, you will obtain an associated four-current density that contains terms relating standard electrical current density. As for your question about special relativity and local gauge invariance. Strictly speaking, special relativity only has a continuous global symmetry, the poincare group, which is made up of the lorentz group (spacetime boosts, spacetime rotations) with the addition of spacetime translations. Jumping back to electricity and magnetism, enforcing local gauge invariance requires that the photon is massless. This is a definitely important for special relativity because the photon is assumed to be as such; only massless particles can keep pace with light. Neat info: because of the link to this gauge symmetry, you can actually experimentally verify charge conservation by measuring a zero mass photon. If the photon were massive, then the gauge symmetry is destroyed and you lose your conserved current. This is why you must have local charge conservation. No local charge conservation => massive photon => speed of light is not an invariant quantity.

Edit: Here is a link that has some information about the lorentz group. I wanted to mention the the four connected subgroups in my original post but didn't want to drone on. From them, you can derive the CPT symmetries and so forth.

http://en.wikipedia.org/wiki/Lorentz_group


Edit 2: Here is my favorite book on the topic of calculus of variations. This theoretical machinery is the foundation for mechanics, and really, your most important tool in theoretical physics. With it, you derive all of the fun contained in Noether's theorem. It is my opinion that no physics student should be without a copy of Weinstock's book.

http://www.amazon.com/Calculus-Variations-Applications-Physics-Engineering/dp/0486630692

Edit 3: Last one, I promise haha. Here is my other favorite, if you are interested in cutting your teeth in a more mathematically rigorous way. Also an excellent book on the topic, it contains a lot the the other book is missing. I want to say that Weinstock doesn't cover the calculation of the second variation(and beyond), which you use to prove that your extremized functional is a minimum or maximum.


http://www.amazon.com/Calculus-Variations-Dover-Books-Mathematics/dp/0486414485/ref=pd_bxgy_b_img_y