Taking a year off before beginning JD/PhD studies. Planning to review/extend my understanding of modern math. Despite being able to successfully and contentfully work problems from Rudin, Folland, etc., I have always struggled to learn from the canonical algebra and geometry books (Dummit, Hatcher, etc.).
The problem doesnt seem to be "local:" there are no specific tricks on particular pages that I can point to that I "dont get." It seems "global:" when reading on analytic topics, thoughts on applications -- if even indirect ones (that is, applications of a collection theories to a more applied branch of mathematics) -- are integral to my efforts; I tend to struggle to find direct and indirect applications of apparent import in algebra and topology.
I believe one way to remedy that fact in learning about algebra proper is to focus on linear algebra and its generalizations: books by hoffman, artin, maclane, etc. seem, after only a cursory look though, much more intriguing to me; it is easier for me to get excited about working the exercises therein. I would like to ask your help in finding suitable perspectives from which to study algebraic topology and geometry, along with reference requests.
I have the suspicion that focusing on the differential aspects of things might be the right direction, but it is tough to find comprehensive advice about what such a curriculum would look like. Based on some cursory reading and comments below, 3 options present themselves so far: spivaks DG1 then narishmans complex analysis, lees manifolds and geometry, or tus manifolds and forms. May I ask your help in comparing those 3 options (coverage? exposition quality? typo quantity?), or recommending additional resources?
In a comment I was a bit more succinct/explicit about what I am looking for: a motivated introduction to a bit of modern geometry, whereby motivated I mean either A) something where I can point directly to an area of mainstream physics or economics and say, "this will be useful here," or B) something where I can point indirectly to an area of mathematics where A obviously holds (differential equations, probability theory, etc.).
From the answers, comments, and elsewhere, it seems that some mix of John Lee and Loring Tu's books have a sufficient smattering of the algebraic side of things to get a feel for modern geometry, with enough grounding in the differential side of things so that the usefulness of the material in applied math is readily apparent. Given the time constraints and the level of difficulty of the available suggestions, algebraic geometry will be omitted (this includes even things like several complex variables by Narishman which discuss the topic tangentially) as will traditional graduate algebraic topology (hatcher, rotman, etc.).