I have recently been told of a proposal to produce an English translation of Landau's Handbuch der Lehre von der Verteilung der Primzahlen, and this prompts me to ask a more general question:
Which foreign-language books would you most like to see translated into English?
These could be classics of historical interest, books you would like your students to read, books you would like to teach from, or books of use in your own research.
Grothendieck's EGA and SGA.
The other two volumes of Kazuya Kato's trilogy on Number Theory (the first vol. is "Fermat's Dream").
Gabriel's dissertation,Serre's FAC and Beilinson-Bernstein-Deligne
Oh my. Since English is already so overwhelming in international scientific literature, I think it will look a bit peculiar to the non-native English speakers who read this site to see a question like this asking for yet more work to be put in English. Perhaps those of us who already speak that language should expend some more effort in the other direction if we want to read something in those other languages.
"La Clef des Songes", "Récoltes et Semailles" and the Long March through Galois Theory.
G.M. Fichtenholz - Analysis (3 Tomes) - The course of real analysis for budding mathematicians beyond the Iron Curtain. Everyone knows it. It's the first book you read, and the last one you refer to before finishing your master's degree. It takes you from the definition of a set to advanced multivariate calculus; it gives you a lot of tools for classical mechanics in the meantime. It is so trustworthy that the single wrong theorem that it contained caused a telltale student to fail his dissertation, because neither he nor his professor checked the proof and they based the whole thesis on the false premise - that was a decade or two ago and the book is, right now, free of errors. Originally in Russian. Another book that kept the Russians strong during the cold war. Wikipedia entry about the author
Hanspeter Kraft's invariant theory book.
As far as I know, none of Wilhelm Blaschke's books have ever been published in English, and he is the the author of possibly the most exciting and elegant serious mathematics books that I've ever encountered (comparable to the best of Felix Klein, but on a much higher mathematical level). I especially regret that his
Einführung in die Differentialgeometrie (1950; 2nd ed with Reichardt, 1960)
and
Elementare Differentialgeometrie (5th edition with Leichweiss, 1973)
have not been available, but really, all his books, from the elementary "Kreis und Kugel" to the state-of-the-art research "Geometrie der Gewebe" are incredible. Fortunately, most of them have been translated into Russian.
Does anyone know a credible explanation of why he was completely ignored in the English-speaking world? Anything to do with WWII? Although even Hasse got translated.
Endliche Gruppen by Huppert, though the German is like the French in EGA: "easy" going.
I'd second Allen Knutson's suggestion that the book Geometrische Methoden in der Invariantentheorie by Hanspeter Kraft (Vieweg, 1984) is a good candidate for translation into English. Since AMS distributes several Vieweg series in English versions, I'd suggest asking Sergei Gelfand at AMS whether such a translation could be commissioned by them.
Like most native users of English, I find mathematical French far easier than mathematical German. In any event, French books and papers are less likely to get translated than German ones. As far as books go, I regret that J.C. Jantzen's useful Springer Ergebnisse volume on primitive ideals in enveloping algebras is not available in English. His later books in English have become standard references for representations of algebraic groups and for quantized enveloping algebras.
Don Zagier's German book about quadratic forms.
Analysis Situs by Poincare.
This is the foundation of algebraic topology and illustrates its historical connection with dynamics.
According to Wikipedia it has been translated, but I can't find a copy in English.
Abel's complete works.
Groupes Algebriques by Demazure and Gabriel. Someone tried to translate the first half of this book, but it's not very good (some of the mathematics is incorrect too).
Gesammelte Abhandlungen (Collected Works) of Carl Ludwig Siegel
(According to Amazon.com a trilingual version of this once existed, but I can't find it.)
"Arithmetique Des Algebres De Quaternions" by MF Vigneras
G. Banaszak, W. Gajda - Elementy Algebry Liniowej (Elements of Linear Algebra), Poland, WNT - 2 tomes - Don't let the name fool you. This recent publication has more linear algebra than you can shake a stick at. It's a very comprehensive course of linear, and some abstract, algebra; very beautifully printed, lots of decorative markup. The book is very well structured, but is not easy and requires the reader to be fully aware of what's going on. It can be a bit of a mind wringer, but on the other hand that can force you to look at many things from the writers' - quite original sometimes - viewpoint. Tome 1 on the publisher's website
This is just a quick round-up of some good books in Mathematics. Hope this helps!
[Another answer contains this suggestion, but it's at the end of the answer and no details are given.]
I would rather like to read Kostrikin's Introduction to Algebra (the 2nd edition, published in 2000: Кострикин – Введение в алгебру). It is in 3 volumes: 'Basic algebra', 'Linear algebra', and 'Fundamental structures of algebra'. Approximately, they cover:
I – preliminaries, matrices & determinants, basics of groups rings & fields, complex & real polynomials
II – vector spaces & linear operators, euclidean hermitian affine & projective spaces, tensors
III – structures of various groups, basic representation theory, rings modules & algebras, Galois theory
The book begins with a discussion about what algebra is, a historical overview, and a set of substantial problems that can be solved with algebra as motivation. Each volume contains a number of figures (67 in total), many applications, and a discussion of open problems (e.g. the convergence of Newton's method, finite projective planes, the inverse Galois problem).
From the Zentralblatt review: "The distinguishing features of the book are the following ones: 1) clearness, clarity and compactness of exposition; 2) the concentric style of presentation; 3) variety of skilfully selected examples (from very simple to very complex ones)."
[Note that the 1st edition was translated, but it is about a third as long and covers far less.]
I.N. Bronstein, K.A. Semyendayev - Mathematics Handbook - an awesome, very complete mathematics handbook for applied mathematicians, physicists, and engineers. Also useful for the pure mathematics researcher who just wants to quickly look up how a basic item in mathematics worked. This work has not lost any of its gleam since it was first written; numerous updates have been made; it is the reference compendium in Central and Eastern Europe. It has received prizes for being the best illustrated engineering book; indeed, the drawings are exact and even beautiful, and have not become outdated in the time of computer generated imagery. Definitely one of the books that put the Russians in outer space. Numerous German editions of the book on Amazon
The following wonderful 54 page survey by O. Neumann on Kronecker's divisor theory could easily be turned into a book and would fill a very large gap in the English literature on such. I'm interested in helping if anyone is game for such (but, alas, my German is weak).
Neumann, O.(D-FSU-MI) 2003k:13021 13F05 (01A55 13G05 20M14)
Was sollen und was sind Divisoren? (German. German summary)
[What are divisors and what can we do with them?]
Math. Semesterber. * 48 (2002), no. 2, 139--192.
In the first part of this paper a survey is given of the development of Kronecker's theory of divisors. In the second part the author develops a theory of integral domains $R$ having a divisor theory in the following sense: there exists a monoid $D$ (i.e., a commutative semigroup with cancellation and a unit element) with the GCD-property for the associated group $G$ of quotients, and a homomorphism $\mathrm{div}$ of the multiplicative group $K^*$ of the quotient field of $R$ into $G$ with the following two properties:
(i) If $a,b \in K^*$ and $b/a \in R$, then $\mathrm{div}(b)/\mathrm{div}(a) \in D$, and
(ii) for every element $d \in D$ there exists a set $A \subseteq K^*$
such that $d$ is the gcd of $\{\mathrm{div}(a) : a \in A\}$.
The author states that a similar theory was presented in the thesis of
F. Lucius ["Ringe mit einer Theorie des groessten gemeinsamen Teilers", Ph.D.
thesis, Univ. Gottingen, Gottingen, 1996; Zbl 0901.13002]. After developing
the fundamental properties of such divisor theory, relations to the approaches
of Kronecker, Zolotarev and Dedekind are established.
--Reviewed by W. Narkiewicz
Équations différentielles à points singuliers réguliers, by Deligne.
Joseph Louis Lagrange - Reflexiones sur la Resolution Algebrique des Equations. I've found lots of discussions and summaries of its contents (e.g. in Harold Edwards' book on Galois theory) and little snippets translated here and there (e.g. in Mathematical Expeditions by Laubenbacher and Pengelley) but haven't been able to locate a complete translation.
Kalkül der abzählenden Geometrie By Hermann Schubert
Lehrbuch der abzählenden Methoden der Geometrie By Hieronymus Georg Zeuthen
[original suggestion/answer by Andrew L]
Constantin Carathéodory's Vorlesungen über Reelle Funktionen. Why this book has never been translated into English is simply a mystery to me.
And while he's at it, let's get whoever's on that case to get Courant and Hurewitz's treatise on complex functions into English as well, so I can see finally if it's as good as Serge Lang always said it was...
2 last requests while I'm at it: Faddeev's 1984 Lectures In Algebra and the second edition of Kostrikin's 3 volume Introduction To Algebra. I'm such a sucker for Russian texts, they're so beautiful and concrete with connections to physics. We Westerners can learn so much from their approach.
B. P. Demidovich - Problems on Multivariate Analysis (approximate translation). A very tough book about analysis on $\mathbb{R}^n$; in fact all problems 'can' be solved by first- or second-year students, but it's got lots of tricky questions that will not let you sleep at night. Only the best need apply - the book gives you the most basic definitions and then throws you out with a broken pontoon in the middle of the ocean, at night. I believe the writer is Russian or Belorussian, I have only encountered a few tattered copies that have been doing the rounds between students for a decade at least. Haven't found a better book for tough multivariate analysis.
Weber's Lehrbuch der algebra, though I see this math underflow post that already expresses this wish: https://math.stackexchange.com/questions/438643/translation-of-webers-lehrbuch-der-algebra-vol-1-2-3
As a textbook bridging from elementary algebra to (then state-of-the-art) research, and coming right at the border between two eras of mathematics, it would be helpful for historians of algebra and mathematics to have this text in english.
I humbly thought I was up for it a year or so ago, but obviously someone more esteemed than I should handle such a monumental translation!
(Edit: "humbly" should read "in ignorance")
Bombieri's "Le Grand Crible dans la Théorie Analytique des Nombres"