Hello all, I'm beginning a self-administered education in math and physics as substitution for a college curriculum, and am looking for some advice on what my general course of study should be. My goal is to learn advanced physics and maths -- I'm most interested in topology and differential geometry, especially as these relate to physics. I have bookmarked probably 50 books on Amazon (fortunately I have the money to spend on whatever books I need), sifted through all the great names in Mathematics (Rudin, Spivak, Munkres, Coxeter, et al) and their works, but I'm slightly at a loss for how I should plan to climb this immense mountain. Right now I have well-absorbed high school knowledge of algebra, geometry and trigonometry, and pre-calculus (geometry was always my favorite... the teachers that could relate more advanced math geometrically made it stick for me). I know that, understandably, Calculus is the default starting point, so I take as a given that I should get very acquainted with Calculus I-II material and perhaps review advanced algebra before starting anything else. What's next? What 'plateaus' or 'levels' are there that are necessary to be completed before moving on to the higher maths? Here's some areas I'm interested in: Linear Algebra Abstract Algebra Differential and Riemannian Geometry Topology / Algebraic Topology Real and Complex Analysis ...I'm sure there are many in between those, but they're slipping my mind. Again, my 'goal' (but I know there is never an end) is to be fluent in the math of topology and advanced geometry, especially of higher dimensions, to augment my physics studies with. I am expecting a long and somewhat tedious path, but I'm ready. Any and all advice is appreciated. :) P.S. I've already found all the most popular online course materials for college-level math classes. I'm more interested in hearing opinions about how best to approach learning the various subjects of Mathematics (i.e. what order), from a purely mathematical perspective.