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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.).

ANSWER:

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.).

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  • $\begingroup$ If you are looking for exercises in algebraic topology then i heartily recommend Jeff Strom's book "Modern Classical Homotopy Theory". $\endgroup$ – Tyrone Aug 22 '17 at 15:12
  • $\begingroup$ I don't yet know any algebraic topology. I am looking for a motivated introduction to the subject, whereby motivated I mean either A) something where I can point to an area of physics or economics and say, "this will be useful here," or B) something where I can point to an area of mathematics where A holds (differential equations, probability theory, etc.). $\endgroup$ – entprise Aug 22 '17 at 15:21
  • $\begingroup$ I apologise that my comment was not more useful. Strom's book offers a heuristic to the subject which is no less abstract, but something that you may find more appealing than the drier material in other texts. You could try Ghrist's book "Elementary Applied Topology" which contains a fair bit of algebraic topology. I'm affraid that I have little experience with this book, but the author claims that "The intent is breadth in ideas, tools, perspectives, and applications...", so maybe it will appeal to your practical outlook. $\endgroup$ – Tyrone Aug 22 '17 at 16:44
  • $\begingroup$ Thanks for the follow up. I have emailed Ghirst; the book seems very interesting, but is unsuitable as a textbook from which to learn those techniques for the first time. It would be like learning differential geometry from amari's statistics book or carroll's relativity book. It seems that Strom is a book of problems, rather than a textbook (read the preface on amazon); looking for something more elementary and complete, than that. $\endgroup$ – entprise Aug 22 '17 at 16:58
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I think you're looking for Introduction to Topological Manifolds by Lee: http://www.springer.com/us/book/9781441979391.

A book that approaches algebraic geometry from a more geometric perspective is Principles of Algebraic Geometry by Griffiths and Harris: http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471050598.html.

Motivation for studying much of this stuff in the context of mathematical physics can be found in Topology, Geometry and Gauge Fields by Naber: http://www.springer.com/gp/book/9781441972538.

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    $\begingroup$ Thanks! The first chapter of Lee is exactly the sort of perspective I was looking for. May I ask though, how do lees books compare with tus? And, frankly, I still do not get from the first chapter of lee why I shouldnt just read spivak -- what about the algebraic topology of things can I not do in differential geometry? It might be that spivaks, lees, or tus books on geometry are the way to go... but how to decide/compare as someone who doesnt yet know any of the material? And, why not play around with a book on several complex variables, where I believe many of these ideas originated? $\endgroup$ – entprise Aug 22 '17 at 21:55
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    $\begingroup$ I think Spivak's books are a little out of date. I actually find Lee's books to be a bit dense/terse, but his first book in the series (the one above) is largely focused on algebraic topology, which you seem to want. Griffith and Harris goes into complex manifolds, which you also seem to want. I find Tu's books to be very very nice to learn from; and they start off on a more elementary level. What one chooses is largely a matter of personal taste, interest, and prerequisite background. You should look into different options and focus on what you take to, understand, and enjoy reading. $\endgroup$ – g.s Aug 22 '17 at 23:18
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    $\begingroup$ Your question "what about the algebraic topology of things can I not do in differential geometry" makes it somewhat clear that you do not know any algebraic topology nor any differential geometry. Why not just pick any sensible textbook and learn them? :-( Being a book sommelier without knowing a subject is not a particularly useful things. $\endgroup$ – Mariano Suárez-Álvarez Aug 22 '17 at 23:45
  • $\begingroup$ If I am understanding you correctly, lees books are faster paced than tus? How does the coverage compare? Simce I am self studying: which has more [comfusing] textual errors? Separately, Griffith might be beyond me for some time yet, but it looks like a great book, Ill keep it in mind down the road! $\endgroup$ – entprise Aug 22 '17 at 23:46
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    $\begingroup$ In my experience, there is little point in these fine discussions on textbooks. To learn a subject, pretty much any book will do. When one knows the basics of a subject, then yes, fine tuning the next book is useful. At that point, the student has some knowledge on the subject and is able to judge. $\endgroup$ – Mariano Suárez-Álvarez Aug 23 '17 at 0:44
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Perhaps:

Adams, Colin Conrad, and Robert David Franzosa. Introduction to Topology: Pure and Applied. No. Sirsi) i9780131848696. Upper Saddle River: Pearson Prentice Hall, 2008. (Book website.)
Top

supplemented by dipping into

Netzer, Tim. "Real Algebraic Geometry and its Applications." arXiv:1606.07284 (2016).

and other similar surveys.

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  • $\begingroup$ The first one, especially, seems interesting. But, from the preface, it appears to emphasize general topology, which I find to be sufficiently motivated by analysis (hence differential equations and probability theory). What would be wonderful would be a sequel dealing with the material typically covered in the second semester of an undergrad topology class or the first semester of a graduate one. But, I doubt such a book exists, which is why I am curious about things like Several Complex Variables... which is perhaps the origin of some of these troublesome constructs? $\endgroup$ – entprise Aug 22 '17 at 19:43
  • $\begingroup$ @entprise: About half the book (Ch.8++) is focused on "applications." Graduate texts are unlikely to emphasize applications. For that you may need specialized literature surveys. Good luck! $\endgroup$ – Joseph O'Rourke Aug 22 '17 at 19:47
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    $\begingroup$ As regards motivation, you might like to look at this article "Modelling and computing homotopy types:I" available at arXiv:1610.07421: the books it refers to give a distinctive view of algebraic topology in their use of groupoids; the more advanced one uses higher groupoids. A web search on "higher structures in maths and physics" shows the general interest. $\endgroup$ – Ronnie Brown Aug 23 '17 at 11:32
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    $\begingroup$ @entprise If you are interested in an algebraic topology book that transitions from undergraduate to graduate, check out mine: Topology Illustrated amazon.com/dp/1495188752. $\endgroup$ – Peter Saveliev Dec 25 '18 at 16:56

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