Thank you for taking the time to write this - people are unlikely to present a more somber take on higher mathematics. Do you know where can I find these Mumford-Lang lecture notes? Their algorithm is based on algebraic geometry methods, specifically cylindrical algebraic decomposition A road map for learning Algebraic Geometry as an undergraduate. The notes are missing a few chapters (in fact, over half the book according to the table of contents). So when you consider that algebraic local ring, you can think that the actual neighbourhood where each function is defined is the complement of some divisor, just like polynomials are defined in the coplement of the divisor at infinity. Even so, I like to have a path to follow before I begin to deviate. New comments cannot be posted and votes cannot be cast, Press J to jump to the feed. Yes, it's a slightly better theorem. I disagree that analysis is necessary, you need the intuition behind it all if you want to understand basic topology and whatnot but you definitely dont need much of the standard techniques associated to analysis to have this intuition. particular that of number theory, the best reference by far is a long typescript by Mumford and Lang which was meant to be a successor to “The Red Book” (Springer Lecture Notes 1358) but which was never finished. 2) Fulton's "Toric Varieties" is also very nice and readable, and will give access to some nice examples (lots of beginners don't seem to know enough explicit examples to work with). A roadmap for a semi-algebraic set S is a curve which has a non-empty and connected intersection with all connected components of S. Fine. There's a huge variety of stuff. I find both accessible and motivated. And now I wish I could edit my last comment, to respond to your edit: Kollar's book is great. MathJax reference. Is there ultimately an "algebraic geometry sucks" phase for every aspiring algebraic geometer, as Harrison suggested on these forums for pure algebra, that only (enormous) persistence can overcome? The books on phase 2 help with perspective but are not strictly prerequisites. An example of a topic that lends itself to this kind of independent study is abelian schemes, where some of the main topics are (with references in parentheses): You may amuse yourself by working out the first topics above over an arbitrary base. What is in some sense wrong with your list is that algebraic geometry includes things like the notion of a local ring. Underlying étale-ish things is a pretty vast generalization of Galois theory. The following seems very relevant to the OP from a historical point of view: a pre-Tohoku roadmap to algebraic topology, presenting itself as a "How to" for "most people", written by someone who thought deeply about classical mathematics as a whole. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. I left my PhD program early out of boredom. DF is also good, but it wasn't fun to learn from. Finally, I wrap things up, and provide a few references and a roadmap on how to continue a study of geometric algebra.. 1.3 Acknowledgements I would suggest adding in Garrity et al's excellent introductory problem book, Algebraic Geometry: A Problem-solving Approach. proof that abelian schemes assemble into an algebraic stack (Mumford. This is is, of course, an enormous topic, but I think it’s an exciting application of the theory, and one worth discussing a bit. real analytic geometry, and R[X] to algebraic geometry. However, I feel it is necessary to precede the reproduction I give below of this 'roadmap' with a modern, cautionary remark, taken literally from http://math.stanford.edu/~conrad/: It is interesting, and indicative of how much knowledge is required in algebraic geometry, that Snapper recommends Weil's 'Foundations' at the end of this "How to get started"-section. Right now, I'm trying to feel my way in the dark for topics that might interest me, that much I admit. ... learning roadmap for algebraic curves. It is interesting, and indicative of how much knowledge is required in algebraic geometry, that Snapper recommends Weil's 'Foundations' at the … Unfortunately I saw no scan on the web. We first fix some notation. Algebraic Geometry seemed like a good bet given its vastness and diversity. Here is the current plan I've laid out: (note, I have only taken some calculus and a little linear algebra, but study some number theory and topology while being mentored by a faculty member), Axler's Linear Algebra Done Right (for a rigorous and formal treatment of linear algebra), Artin's Algebra and Allan Clark's Elements of Abstract Algebra (I may pick up D&F as a reference at a later stage), Rudin's Principles of Mathematical Analysis (/u/GenericMadScientist), Ideals, Varieties and Algorithms by Cox, Little, and O'Shea (thanks /u/crystal__math for the advice to move it to phase, Garrity et al, Algebraic Geometry: A Problem-solving Approach. And it can be an extremely isolating and boring subject. Take some time to develop an organic view of the subject. For a small sample of topics (concrete descent, group schemes, algebraic spaces and bunch of other odd ones) somewhere in between SGA and EGA (in both style and subject), I definitely found the book 'Néron Models' by Bosch, Lütkebohmert and Raynaud a nice read, with lots and lots of references too. Then they remove the hypothesis that the derivative is continuous, and still prove that there is a number x so that g'(x) = (g(b)-g(a))/(b-a). Authors: Saugata Basu, Marie-Francoise Roy (Submitted on 14 May 2013 , last revised 8 Oct 2016 (this version, v6)) Abstract: Let $\mathrm{R}$ be a real closed field, and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. Seemed like a good bet given its vastness and diversity is based on algebraic seemed... People are unlikely to present a more algebraic geometry roadmap take on higher mathematics not be cast, Press J jump. Your edit: Kollar 's book is great and it can be an isolating... 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