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Chicago undergraduate mathematics bibliography
Presumably, this was a list of books on Amazon. Any chance we can recover this list?
I would like to recommend a widely-used in Easter Europe and Russian, unfortunately almost unheard of elsewhere, "Differential and Integral Calculus" by G.M. Fichtenholz to intermediate calculus section. First volume covers and supports with multiple examples for each section preliminary material about maps, relations, continuity, differentiation and analysis of function for single and multivariable cases with addendum covering Lagrange multipliers and similar methods. I can describe all volumes in detail if it is needed of me.
In intermediate differential geometry, or possibly calculus, I would like to recommend Loring W. Tu "Introduction to Manifolds" (second edition! there were few nasty bugs initially). Personally and after asking some of my students I find it as more verbose but much more approachable than Spivak's "Calculus on Manifolds". S. Morita "Geometry of Differential Forms" (again, second edition. but mostly because it's the only one I have access to) also would be my pick. It is very light and encouraging application and constant checking of reader's intuition, aside of few spikes it maintains a very gentle learning curve.
I've noticed you've left out Allen Hatcher's Algebraic Topology from your list of textbooks in that topic. I would send a pull request outlining the book, but I've only started reading it, and it might take me a while. I suggest if you've read it, might add an entry for it.
The link in the readme with the label The syllabus of the Cambridge Mathematical Tripos (PDF)
is broken.
First, I would like to thank you for maintaining this list of references--I have found it quite useful to guide me to self-study in areas I didn't have much exposure to as an undergrad at U.C. Berkeley.
Additionally, I would like to make some suggestions with respect to Measure Theory and Probability Theory (will include the latter in a separate item)--while I have not read all of these (so perhaps I cannot provide a full reference), I can indicate those I haven't and perhaps someone else can attest to them:
First, while it was probably true when you first compiled your references that "nobody writes books entitled Measure Theory any more", that appears to no longer be the case. My full year undergraduate analysis sequence included Lebesgue Integration, but via the Riesz method (not measure theoretic) which has required me to do some "catch up" measure theory reading with respect to probability theory. There are two "Intermediate" texts I would like to call out:
Rene Schilling's "Measures, Integrals & Martingales" (have been working through both editions, and the 2nd is definitely an improvement over the already excellent 1st)--I cannot provide enough accolades for this book; the author provides a great deal of motivation to what has historically been a dry subject, demonstrates great finesse in discussing topics in simplifying detail without adopting a "prose" style, and has augmented (not replaced) notation to include inline references to make the proofs and examples crystal clear (as well as indicated alternate notation which is sometimes used but not adopted by the author). I wish this text were around for my second semester undergrad analysis class and we had used it; there is also a PDF of all exercise solutions linked from his home page which I have also found valuable for self-study.
David Cohn's "Measure Theory" (going to move on to this one next; it is definitely at a quicker pace and covers topics such as Polish Spaces in much greater depth). But from my scanning through it, still does seem within grasp of "advanced undergraduates" as the preface suggests, and there is also a review published in the MAA in 2014 which attests to the gaps the volume fills compared with the slick/brief coverage provided by Rudin, or how it is "cleaner and tighter" than Royden in this area.
Lastly, slightly over 10 years ago a 2 volume, 1000+ page work entitled (you guessed it) "Measure Theory" was published by V.I. Bogachev, and while it's been checked out at our library by a researcher (so I haven't had the opportunity to look through it yet), all reviews I have read indicate it is to measure theory what Jech is to set theory--a complete reference to practically any and all findings in measure theory, historical and current, including more than 800 exercises and a bibliography with 2000+ (!) references. If the previous books can serve to make this topic accessible at an intermediate level, this set would definitely appear at a step above (advanced level).
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