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I received an email from zbMATH today to the effect that, as a result of their going open access in January 2021 (which is awesome), they will no longer be able to offer their reviewer discount for Springer books since their distribution contract with Springer is being terminated. They advise that all reviewers use any remaining reviewer credits before January 2021, when the credits will expire.

Edit: As David White suggests in his answer below, some community members may find it preferable to donate their extra credits to mathematicians and departments in places where there is a need for books. One possible way to do this is provided by zbMATH and outlined at the bottom of their guide for reviewers as pointed out by Najib Idrissi in the comments on David's answer, allowing reviewers to donate spare credits to the EMS Committee for Developing Countries, however I have been unable to follow the link provided in the guide to see how promptly they use these donated credits (it won't load for me).

Although I've decided to follow Davids suggestion, I'm leaving up the rest of the question since others have contributed to it.

I thought it would be valuable to collect together some community recommendations to spend reviewer credits on before January 2021 since many community members here are likely reviewers for zbMATH as well.

What Springer book(s) would you recommend spending spare reviewer credits on?

I'm a fan of mathematics in general so any recommendations are welcome, but if I had to choose specific topics I'd love a good reference book on algebraic geometry, homotopy theory, combinatorics or synthetic differential geometry.

Second edit: Since the community decided to close the question, I'm going to leave it up and closed until January then delete it; there's no evident way to make the topic less subjective, and it will be irrelevant after the credits are expired. Until then, this post can still hopefully serve as a resource for people who want to donate or spend their expiring credits.

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    $\begingroup$ I'm not a homotopy theorist, but Fomenko and Fuch's "Homotopical Topology" has some cool illustrations. $\endgroup$
    – RBega2
    Commented Oct 13, 2020 at 22:13
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    $\begingroup$ Given the size of Springer's catalog, this is of the same order of magnitude of breadth as just "What are the best mathematics books"? $\endgroup$
    – Nate Eldredge
    Commented Oct 13, 2020 at 22:53
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    $\begingroup$ @NateEldredge Which is why I would normally not ask a question this broad, but the additional imperative of expiring credits specific to Springer books made me think it was appropriate. If you have a suggestion for how to usefully narrow the question down I'll edit, but I was hoping for some nice 'canonical' Springer suggestions; for example, in a McGraw-Hill book list something like Real and Complex analysis by Rudin would be a 'canonical' book that everyone should see at least once IMO. $\endgroup$
    – Alec Rhea
    Commented Oct 13, 2020 at 22:58
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    $\begingroup$ 2 classics by VI Arnold: on classical mechanics and ODEs respectively. $\endgroup$
    – Piyush Grover
    Commented Oct 14, 2020 at 0:28
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    $\begingroup$ In my modest oppinion, Beck and Sinai's book "Computing the continuous discretely" is a true gem, and a must have for any combinatorialist. $\endgroup$
    – Luis Ferroni
    Commented Oct 14, 2020 at 10:30

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I, too, received the email that the reviewer credits would be expiring. Perhaps this would be a good time to set up a system by which reviewer credits could be transferred to folks with a larger need? I really don't need any more books, but I can imagine that there may be mathematicians and departments in developing countries where these credits might be put to good use. If others agree, and think a large number of reviewers might be willing to give away their credits to those with greater need, maybe we could brainstorm (in chat?) about how to make this happen.

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    $\begingroup$ I would be on board for this. $\endgroup$
    – Alec Rhea
    Commented Oct 13, 2020 at 23:33
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    $\begingroup$ You can already opt to gift your royalties to the EMS Committee for Developing Countries. zbmath.org/reviewer-service/info_texts/guide_for_reviewers $\endgroup$
    – Najib Idrissi
    Commented Oct 14, 2020 at 5:06
  • $\begingroup$ Najib's suggestion sounds perfect, to me. That's probably what I'll opt for (asap, so that the points can still be used). Unless someone else posts a comment with a different idea. $\endgroup$
    – David White
    Commented Oct 14, 2020 at 10:41
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    $\begingroup$ As noted above, the EMS website seems to be down today, but the Website of the EMS Committee on Developing Countries is available at ems-cdc.org, with its activities at listed at nickpgill.github.io/emscdc/activities, including the book donation program. $\endgroup$
    – Olaf Teschke
    Commented Oct 14, 2020 at 16:25
  • $\begingroup$ We have SciHub and Genlib so we're fine really $\endgroup$
    – Marko Karbevski
    Commented Oct 16, 2020 at 5:17
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M. Aigner, G. Ziegler, Proofs from THE BOOK.

This is a book that every mathematician should have. From the reviews:

"Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... "

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Bott and Tu's "Differential Forms in Algebraic Topology" is probably the best-written math textbook I've come across, and the material covered there is substiantially different from that in, say, Hatcher or Spanier.

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Although they are basic, I think the books by Lee on topological manifolds, smooth manifolds, and Riemannian manifolds should be on such a list if we are talking about which books are most generally used by the most people.

They continue to be extremely useful to countless mathematics students and people who are self-studying mathematics including physicists who need to learn more about manifolds.

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Anything by Serre.

I particularly admire Trees. One has to get past the formalism, but then one gets an incredibly concise exposition of many powerful theorems. It feels like every word was carefully weighed (even when read in translation). For instance an immediate corollary of the result that a group is free if and only if it acts freely on a tree is that subgroups of free groups are free.

Another remarkable Serre book is Linear representations of finite groups, which has three parts, at wildly different levels. The final part is still an excellent reference for the decomposition map relating group representations in zero and prime characteristic.

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    $\begingroup$ Which one ? Jean-Pierre ? $\endgroup$
    – Denis Serre
    Commented Oct 14, 2020 at 17:03
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    $\begingroup$ I apologise. Please may I also recommend the lovely book Matrices: Theory and Applications, by one Denis Serre, which I have often found useful as a reference. $\endgroup$
    – Mark Wildon
    Commented Oct 14, 2020 at 17:43
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    $\begingroup$ Never mind, I am proud of my uncle ! $\endgroup$
    – Denis Serre
    Commented Oct 14, 2020 at 18:17
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    $\begingroup$ @DenisSerre And jealous too :) $\endgroup$
    – Piotr Hajlasz
    Commented Oct 15, 2020 at 22:50
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    $\begingroup$ Anything by anyserre. $\endgroup$
    – Simon Pepin Lehalleur
    Commented Oct 16, 2020 at 5:56
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Since you mention algebraic geometry, here are some Springer books that I use frequently:

  • Algebraic Geometry: Hartshorne, Robin
  • Geometric Invariant Theory: Mumford, David, Fogarty, John, Kirwan, Frances
  • Geometry of Algebraic Curves Volume I: Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.D.
  • Positivity in Algebraic Geometry I & II: Lazarsfeld, R.K.
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    $\begingroup$ In more specific areas of algebraic geometry than Hartshorne, I'm personally very fond of the Springer books "The Arithmetic of Elliptic Curves" and "Advanced Topics in the Arithmetic of Elliptic Curves." :) $\endgroup$
    – anomaly
    Commented Oct 14, 2020 at 14:32
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    $\begingroup$ Is Hartshorne still relevant nowadays? I thought every single thing in it has been explained better somewhere else by now, even though "somewhere else" may be different places depending on the thing. I'd certainly include Eisenbud's With-a-view in the list, though... $\endgroup$
    – darij grinberg
    Commented Oct 14, 2020 at 15:12
  • $\begingroup$ @darijgrinberg It is certainly still used a lot. As you can explore on mathscinet, it is already for many years by far the most-cited math book outside of analysis. $\endgroup$
    – Lennart Meier
    Commented Oct 14, 2020 at 15:27
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    $\begingroup$ @anomaly Thanks. I, too, am fond of those books. :) $\endgroup$
    – Joe Silverman
    Commented Oct 14, 2020 at 17:18
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    $\begingroup$ @anomaly: And let us not forget, for completeness at least, the undergraduate Springer title "Rational Points on Elliptic Curves" - also a book to be fond of. (Doubtless, one if not both of the two authors will agree, and rightly so.) $\endgroup$
    – J W
    Commented Oct 16, 2020 at 20:29
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Polynomials and Polynomial Inequalities, Peter Borwein & Tamás Erdélyi (GTM 161). There's something for everyone in there.

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Of my modest library, Neukirch's Algebraic Number Theory is easily my favorite. It's approach to class field theory avoids cohomology, so a student without a heavy algebra background can use it as a second course in algebraic number theory. I appreciate the third chapter as a down-to-Earth glimpse of Arakelov theory as well. Certain sections, exercises, and remarks hint at deep connections between number theory and algebraic K-theory. And of course Neukirch's style and organization is simply delightful.

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  • $\begingroup$ For german speaking persons, I believe it is also available in its original language. $\endgroup$
    – François Brunault
    Commented Oct 18, 2020 at 20:43
  • $\begingroup$ Sometimes I wish I spoke German for this very reason. $\endgroup$
    – Nico
    Commented Oct 18, 2020 at 21:10
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    $\begingroup$ Laurent Schwartz used to say that reading math in a foreign language is a good way to learn this language. $\endgroup$
    – François Brunault
    Commented Oct 18, 2020 at 21:12
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I am by no means an expert but I really like Introduction to Number Theory by Hua Loo Keng (L.-K. Hua) because of its broad coverage of not just the standard topics in a generic number theory text, but also including chapters such as Schnirelmann density and geometry of numbers.

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The question depends so much in your field of interest that it is impossible to give an exhausting answer. Some of my favorites in various fields are the classics

  • Nicolas Bourbaki, General Topology
  • Glen E. Bredon, Topology and Geometry
  • Shui-Nee Chow and Jack K. Hale, Methods of Bifurcation Theory
  • Klaus Deimling, Nonlinear Analysis
  • Albrecht Dold, Lectures on Algebraic Topology
  • Herbert Federer, Geometric Measure Theory
  • Morris W. Hirsch, Differential Topology
  • Thomas J. Jech, Set Theory
  • Tosio Kato, Perturbation Theory for Linear Operators
  • Mark A. Krasnoselskij and Petr P. Zabrejko, Geometrical Methods of Nonlinear Analysis
  • Joram Lindenstrauss and Lior Tzafriri, Classical Banach Spaces (2 volumes)
  • Jacques-Louis Lions and Enrico Magenes, Non-Homogeneous Boundary Value Problems and Applications
  • George W. Whitehead, Elements of Homotopy Theory
  • Eberhard Zeidler, Nonlinear Functional Analysis and Applications (several volumes)

Then there are of course a lot of the Lecture Notes and also some very good books in German.

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