Sets and Topology for Physicists?
A project that has been on my mind for awhile with regards to books and literature that go, "Insert math field here for Physicists and Engineers," is that what about set theory and topology for physicists? Well, there is one by Nash and Sen (here) and one of my favorite, yet seemingly quite difficult, textbooks called The Geometry of Physics by Frankel (also here). The first text is an overview and an outline of how to calculate various physical quantities in terms of their full topology while the second goes into how every single aspect of physics can be built from geometry (awesome).
However, I have a few issues though which each; the Sen and Nash text gives (what appears to be) a very basic overview of what topology is and assumes knowledge of set theory and the foundations of mathematics (courses like discrete & foundational mathematics), and doesn't include any problems or exercises. The text, as stated, just includes, "here is the quantity defined, and here is where we use it and how." My issue with the Frankel text is that it is very heavy on the mathematical language and precision, which is amazing but also terrifying for someone entering the field of geometrical physics. Now, I have gone through the textbook Gravitation and have gone over quite a few problems in the textbook; but it is the style of how it introduces the mathematical language starting from: differential forms, then differential topology, then Riemannian geometry. And along each step and introduction of a new topic is a "story," a tale of how we as physical creatures, can travel along geodesics and measure their curvature via these cool machines called tensors. This is non-existent in Frankel's text. But, where Gravitation misses out on some cool physics, Frankel gives it to us. Also, Frankel has a lot of nice problems.
So, it would seem then that my goal, or I suppose learning challenge, is to write-up/teach myself set theory for physics, but in my own way: taking the formalism of Frankel + Sen & Nash and tying it together with Gravitation. But, write-in what both text are lacking, an introduction to sets and mathematical foundations. This way, someone who, lets say, is a third year in college on the physics major path, has just realized he wants to learn about geometrical physics. But, they have never taken any mathematical courses outside of the basic calculus path, so where will they turn to? Their schedule is loaded with a nice portion of credits, they are a tutor plus LA/TA, conducting research, but still has this burning passion to learn about geometrical physics. Well, this is the niche I want to fill.
The plan then that I have in mind is to learn/study along the following path. First, Mendelson has a nice (and cheap) text on topology provided by Dover which seems to be a nice home to learn the formalism of sets and topology. Armed with this formalism, revisiting Sen and Nash's text will be the proper next step, since they include relevant topics such as homology in quantum field theory, and Yang-Mills. I am not sure if I will include anything from Frankel, since that is a nice text built on Riemannian geometry (which is also in every gravitation text as well). However, what is usually not included in gravitational texts, which we will include is Penrose's work on differential topology and relativity. This will include his work on singularity theorems and causal sets (since they appear to be important for quantum gravity). I may also include a section on symplectic geometry and where it is being used, since this subject does not get a lot of attention.
It should be noted that this is months out. I may be able to start this project once I finish LQG and studying for the GRE; so expect an update on this around November. Either way, if you have thoughts or directions on this, don't be afraid to send them in.
Have a great weekend.