Analytic Geometry Prerequisites
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I am a 3rd undergrad student majoring in Algebra and Topology. According to you, what is the ideal background I should have before diving into Rigid Geometry? What are some great resources (books, videos,..) about this topic that you would prefer to me?
A bit about my current knowledge: Commutative Algebra up to Michale Atiyah, Algebraic Topology up to Hatcher, Fields and Galois theory up to Steven Roman, Manifolds (I haven't learned anything about smooth manifolds, I just finished Topological Manifold by John Lee), a bit about Homological Algebra and nothing about Algebraic Geometry.
Now while there is nothing stopping you from jumping right into these notes and trying to digest the material, I would add a small caveat.Just as you can immediately start learning about algebra (field extensions, Galois theory...) without learning linear algebra before, you can also learn about rigid geometry without prior knowledge of algebraic geometry.
The problem that you might be facing is that often things might seem unmotivated, lacking intuition and difficult to understand. Things like sheaf theory are important, if not concretely, then at the very least on a conceptual level. These ideas came to life initially in algebraic geometry, when people introduced schemes, tale morphisms/topology etc. To a certain degree rigid geometry (whatever that is supposed to mean) was developed after algebraic geometry, in parts it aims to mimic it.
Thus let me suggest that it is very helpful to first learn about algebraic geometry (along the lines of Hartshorne Chapter II or Gtz and Wedhorn's Algebraic geometry and other possible sources) and also something about non-archimedean fields/algebraic number theory (along the lines of Serre's Local fields). Certainly there is no need to learn everything up to the last detail, but working with these objects will give you an indispensible intuition.
Afterwards you can think, if you first want to read about rigid-analytic spaces la Tate (Bosch's Lectures on Formal and Rigid Geometry is a great resource here) or jump right into action in the aforementioned source. I personally did the latter and didn't find it to be such a big hassle. A great supplement to the Berkeley notes are the notes from this Number theory seminar. You can of course also read about perfectoid spaces straight from the source in the survey paper, but I found this to be more technical. In any case it is a great back-up source. Also you can find more references in these study group notes.
Perfectoid spaces are this year's subject for the Arizona Winter School (link) and, as preparation, I am currently trying to understand the subject better. There are wonderful explanatory accounts (What is a perfectoid space, What are "perfectoid spaces"?) but I would like to learn the technical details. An additional problem would be that I don't know much (anything) about rigid analytic geometry. I have found many lecture notes on the internet treating non-archimedean geometry, adic-spaces, Berkovich spaces etc but I don't know where to start learning. Moreover, I don't know which things are important and which things can be skipped at first reading. I am familiar with schemes and local fields at the least.
I can tell from personal experience that it is possible to learn perfectoid spaces without knowing rigid geometry, just like it is possible to learn schemes or even stacks without knowing much about varieties over complex numbers. In fact, it's even possible to successfully transition to research with this approach. Of course, for the approach to be meaningful/successful you need some "mathematical maturity" (in the sense of being able to clearly distinguish easy/formal parts of the theory from the real meat); for instance, you need a good command of commutative algebra/algebraic geometry (to say the least).
I do agree with the others though that it may be a little bit too late for the AWS to be entirely meaningful. Even if you do not completely understand the lectures, try to isolate the contact points with the material that you've been studying in order to get something out of them.
I would begin with Brian Conrad's "Several approaches to Non-Archimedean geometry" chapter in a set of lecture notes of a previous AWS -getitem?mr=MR2482345 and there are many useful exercises there. Also see the second half of Tate's foreword and Berkovich's foreword to this book for quite an enlightening historical perspective.
For all the details in say a concrete case like the rigid projective line there is an e.g. first chapter in excellent textbook of Fresnel and van der Put -getitem?mr=MR2014891 and the rest of the book has a lot of things that are difficult to find elsewhere. Note this isn't just a translation of the French version, there is a lot more in it than in the 1981 book.
Then I guess for the adic case which isn't such a massive jump from Berkovich spaces, there are excellent lecture notes of Wedhorn (his webpage) and also its worth noting Huber was an extremely clear writer of articles, and for the perfectoid theory well you are lucky because Scholze is also a remarkably clear writer.
The teaching of calculus in the high school has been an issue quite heavily debated for the last few years. Educators, mathematicians, and textbook writers have written much concerning how and when to teach calculus. However, few people have written concerning the prerequisites necessary for calculus. The basic assumption underlying this study was that there would be advantages present if a student could have a thorough understanding of analytic geometry before he encounters a calculus course. This claim is made because; first, the student would probably be able to concentrate on the calculus itself and not be slowed by a lack of knowledge about analytic geometry. Second, since most calculus courses include some analytic geometry, some analytic geometry topics would be presented to the student for a second time. The primary objectives of this study were 1) to determine those specific topics of analytic geometry which are recommended as essential prerequisites to calculus; 2) to identify topics taught in analytic geometry which are contained in ten selected high school mathematics text books and 3) to determine if a need exists for teaching analytic geometry as a prerequisite to calculus. In reviewing the literature, the investigator located some published material which indicated a need for teaching analytic geometry as a prerequisite to calculus. A criterion list was developed by reviewing the recommendations of the School Mathematics Study Group (SMSG), University of Illinois Committee on School Mathematics (UICSM), and the Committee on the Undergraduate Program in Mathematics (CUPM), and three college calculus textbooks. The topics on the criterion list should exemplify 1) the recommendations of the mathematics study groups and 2) content of the college level analytic geometry textbook. The criterion list of analytic geometry topics was then compared with ten state-approved twelfth grade textbooks. The evaluation of the textbooks consists of finding the prescribed topics and then determining how many pages were allotted to the topic in each book. It would seem that the topic on the criterion list were important for most of the ten textbooks had some pages allotted to the discussion of each topic. No attempt was made by the investigator to determine which textbooks were better or which ones contained more material concerning analytic geometry. All the investigation attempted to do was to identify those topics in analytic geometry which were important as prerequisites to calculus and then compare this list with certain twelfth grade textbooks.
Geometric congruence, similarity, area, surface area, volume, introductory trigonometry; emphasis on logical reasoning skills and the solution of applied problems. This course may not be used to satisfy the basic minimum requirements for graduation in any baccalaureate degree program.
Natural numbers; integers; rational numbers; decimals; ratio, proportion; percent; graphs; applications. Students who have passed MATH 001 may not schedule this course for credit. This course may not be used to satisfy the basic minimum requirements for graduation in any baccalaureate degree program.
Algebraic expressions; linear, absolute value equations and inequalities; lines; systems of linear equations; integral exponents; polynomials; factoring. This course may not be used to satisfy the basic minimum requirements for graduation in any baccalaureate degree program.
This course satisfies the General Education Qualification. Topics covered include visualizing and graphing data; evaluating the average rate of change; solving linear equations and inequalities; solving linear absolute value equations and inequalities; modeling with linear functions and discussion of interpolation; solving quadratic equations using different solution methods; solving quadratic inequalities; modeling with quadratic functions.
This course covers topics that include functions and their representations; distinguishing between types of functions; evaluating the average rate of change; factoring polynomials of general degree; solving polynomial inequalities; solving rational equations and inequalities; solving radical equations; modeling with polynomial and rational functions; finding and interpreting the meaning of inverse functions; solving exponential and logarithmic equations; modeling with exponential and logarithmic functions.
This course satisfies the General Education Qualification. Topics covered include angles and their measures; right triangle trigonometry; all six trigonometric functions and their representations; angle addition/subtraction and double angle identities; modeling with sine and cosine; applications of trigonometric functions; simple harmonic motions and other applications of trigonometric functions; inverse trigonometric functions; solving trigonometric equations; verifying identities; law of sines and law of cosines; vectors; polar equations; trigonometric form of complex numbers; other related topics as time permits.
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