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This has been inspired by this MO question: Harmonic maps into compact Lie groups

Just for joking: which is your favourite never appeared forthcoming paper?

(do not hesitate to close this question if unappropriate)

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closed as no longer relevant by Benjamin Steinberg, Vidit Nanda, Todd Trimble, quid, Suvrit Mar 9 '13 at 1:14

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Community Wiki? –  Dirk Dec 6 '10 at 20:02
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I wonder if Bill Thurston is reading this. ;) –  Nikita Sidorov Dec 6 '10 at 20:17
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@Nikita: Thurston explained his time-investment choices in a remarkable, penetrative essay, On Proof and Progress in Mathematics: arxiv.org/abs/math/9404236 . –  Joseph O'Rourke Dec 7 '10 at 12:13
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All of my papers never appeared! –  drbobmeister Dec 8 '10 at 9:38
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I recalled Robert Thomason citing the various preprints of his paper "Algebraic K-theory and etale cohomology" as "to disappear" We are all glad that the paper finally appeared in Ann. Sci. Ecole Norm. Sup. (1985). –  F Zaldivar Mar 9 '11 at 17:22

26 Answers 26

up vote 57 down vote accepted

This doesn't exactly count as an unpublished forthcoming paper, but the supposed original proof of Fermat's Last Theorem that was "too large to fit in the margin" should probably be mentioned here.

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EGA, Chapters 5 through 12

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I can't believe this doesn't get more votes. –  YBL Jan 25 '11 at 13:32
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Well, the author is on vacation isn't he? :) –  Koundinya Vajjha Mar 18 '11 at 12:37

Volumes 4 through 7 of The Art Of Computer Programming.

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4a is about to come out! –  Daniel Litt Dec 7 '10 at 4:57
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According to Knuth, 4A was completed and sent to the printer today (December 6th, 2010) –  Adrian Petrescu Dec 7 '10 at 6:24
    
@Adrian: Were you at the same talk I was? –  Daniel Litt Dec 18 '10 at 23:20
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I hope he doesn't have a lot of errors in it. :) –  Koundinya Vajjha Mar 9 '11 at 12:14

The comment about stacks in the paper that first used them in an essential way probably belongs in this list:

"Full details on the basic properties and theorems for algebraic stacks will be given elsewhere." (Deligne-Mumford, The irreducibility of the space of curves of given genus, 1969.)

They don't quite say they will give the details in a paper, of course, so maybe it doesn't count.

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I think it counts. Especially since the corresponding book is still forth-coming. –  Sándor Kovács Dec 7 '10 at 6:45

Steven Krantz tells the following story, Mathematical Apochrypha, page 136:

My Ph.D. thesis was based in part on work of Walter Koppelman of the University of Pennsylvania. My source was a very brief research announcement that Koppelman had published in the Bulletin of the AMS. I could never find the promised subsequent paper that would fill in all the details, and I had to fill them in myself. I eventually went to my thesis advisor and asked him where the missing paper was. He said, "Oh, God. Don't you know?" And then he told me the sad story. There was a very unhappy graduate student at the University of Pennsylvania. He had had bad experiences with several thesis advisors (at least so he thought), the last being Koppelman. One day he went into the colloquium, shot the department chairman, shot Koppelman, and shot himself. Koppelman and the student died.

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I can confirm this story. My father was the graduate chairman at the time. He would have been at the talk, but had a headache that day and stayed home. The department chair (I don't remember exactly who, but I think it was Oscar Goldman) was wounded. –  Deane Yang Jan 25 '11 at 3:09
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Gromov's seminal "Pseudo holomorphic curves in symplectic manifolds" (1985) refers 10 or 15 times (for explanations of further applications that he only refers to or sketches briefly and for even "further discussion on $\overline{\partial}_\nu$ for non-regular curves") to his forthcoming "Pseudo holomorphic curves in symplectic manifolds, II", listed as "in press" by Springer.

It never appeared. Gromov wrote a few later papers on symplectic geometry, but never returned to holomorphic curves. The paper is the foundation of modern symplectic topology (Floer homology, quantum cohomology, Gromov-Witten theory, symplectic field theory, etc.)

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Dana Scott and Robert Solovay, "Boolean-valued models of set theory"

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Absolutely — a classic of the literature! –  Peter LeFanu Lumsdaine Dec 6 '10 at 20:02
    
Scott did publish a paper - "A proof of the independence of the continuum hypothesis" (Mathematical Systems Theory, vol. 1, iss. 2, 1967) - on Boolean models and forcing, but the treatment was fairly low-level. I suspect that the intended content of the Scott/Solovay paper is closer to that of the paper "Boolean-valued set theory and forcing" (Synthese, vol. 33, no. 1, 1976) by Richard Mansfield and John Dawson, based off of notes from a seminar run by Dana Scott. –  Noah S Mar 9 '13 at 22:31
    
@Noah: If I remember correctly this paper of Dana Scott presented Boolean-valued models in the context of second or third order arithmetic, concentrating on the particular Boolean extension obtained by adjoining a lot of random reals. (Or am I remembering a different paper from the one you cited.) I agree that the Dawson-Mansfield paper is a reasonable approximation to the nonexistent Scott-Solovay paper. A more detailed version would, I think, be Bell's book. –  Andreas Blass Mar 10 '13 at 2:18
    
@Andreas, yes, that's correct. I was just mentioning that paper because one could run across the date, title, and author, and suspect that it was basically the same as the promised Scott/Solovay, which it is not. –  Noah S Mar 10 '13 at 2:49

Nobody can compete with Fermat, but papers confidently labelled with the roman numeral I and never followed by II might fit here. Of these my favorite is one by Tits, Normalisateurs de tores I in J. Algebra 4 (1966).

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Here's a question: is there a paper labeled with roman numeral II for which part I never appeared? –  Victor Miller Dec 6 '10 at 22:44
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This isn't as dramatic as what you asked for, but you might be amused by looking at the authors of these papers front.math.ucdavis.edu/… –  David Speyer Dec 7 '10 at 3:32
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@Victor Miller: One can ask even further questions. If there are I and III of something but not II? More generally, $n-1$ and $n+1$ but not $n$? –  zhoraster Dec 7 '10 at 7:17
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According to MathSciNet, Antony Wassermann has a series of papers on "Operator Algebras and Conformal Field Theory", with the first appearing in the 1994 ICM proceedings, the third appearing in Inventiones in 1998, and the second .... ? –  Yemon Choi Dec 7 '10 at 17:48
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Well, I got quite a few queries about where part I of "Applications of random sampling in computational geometry II," by Clarkson and Shor, was. (The answer is that it was called "New applications of random sampling ..." and was single-author Clarkson). –  Peter Shor Dec 8 '10 at 20:44

I'm a fan of Peter May's book The Homotopical Foundation of Algebraic Topology (feel free to correct the title if I've got it wrong). It has been referred to by May in various places, and sounds really interesting! But it has never been written.

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I recently emailed him about this and he said that he was commissioned to do this well before I was born, but it will never be written. :) –  Dylan Wilson Dec 8 '10 at 16:16
    
Too bad. His article in Peter Hilton's birthday volume is sort of a preview for the book. It's called The Dual Whitehead Theorems, I believe. –  Dan Ramras Dec 8 '10 at 17:08
    
There is also Concise 2 that is in the works and hidden somewhere on his website. –  Sean Tilson Jan 25 '11 at 5:11
    
That's exciting! –  Dan Ramras Jan 25 '11 at 18:56

The sequel to Kontsevich's "Deformation quantization of Poisson manifolds, I" has never appeared.

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How about "The classification of finite quasithin groups" by G. Mason from 1980? The classification of finite simple groups was announced when G. Mason was still working on this important case and he then abandoned the work. This hole in the classification was closed finally in 2004 by M. Aschbacher and S. D. Smith.

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Deligne's construction of the Galois representations attached to modular eigenforms (he did give a sketch in a Bourbaki talk though).

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And indeed, to this day, I don't think there is a published reference for this result. –  Olivier Dec 7 '10 at 15:32
    
It's been quite a while since I looked at it, but I recall Tony Scholl's article "Motive for modular forms" filling in some of the details of Deligne's construction. –  Ramsey Jan 25 '11 at 2:13
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Of course, I should seek greater pluralism in my motifs. –  Ramsey Jan 25 '11 at 2:30
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Dear Nick, You are correct re. Scholl's article. Dear Laurent, My memory is that Langlands's article in LNM 349 (vol. II of Antwerp) gives the construction (and quite a bit more). Best wishes, Matt –  Emerton Jan 25 '11 at 7:03
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@Chandan: Kevin Buzzard has explained elsewhere on MO that Brian Conrad's book "achieved infinite length." –  David Hansen Mar 9 '11 at 22:20

The books Classical Banach Spaces III and Classical Banach Spaces IV by Joram Lindenstrauss and Lior Tzafriri never appeared (after having been promised in various places of volumes I and II).

As written by Albrecht Pietsch in his book History of Banach Spaces and Linear Operators, the reason the later volumes never appeared was that "the development was too vigorous. Thus, in order to finish this project, a complete rewriting would have been necessary". Even still, the influence of volumes I and II in Banach space theory has been exceedingly nontrivial; indeed, Pietsch also writes: "The two-volume treatise of Lindenstrauss/Tzafriri on Classical Banach Spaces has become the most important reference of the modern period".

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Weren't there actual proofs of results stated in I & II that were meant to appear in III & IV? –  Yemon Choi Dec 7 '10 at 3:24
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@Yemon: At least one such case occurs in the proof that $\ell_\infty$ is prime (Vol. I, p.57). Another instance occurs on p.106. –  Philip Brooker Dec 7 '10 at 3:48
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Although this is off-topic, this thread has reminded me of the fact that the papers Complementably universal Banach spaces and Complementably universal Banach spaces II by Johnson and Szankowski appeared in print some 33 years apart. I wonder if there is a bigger gap between a paper and its sequel? –  Philip Brooker Dec 7 '10 at 5:11
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If books are allowed, there will be a ton of examples. Because it is much more difficult to put an end mark to a book than to an article. Let me give two items. The book on semi-groups by Benilan, Crandall and Pazy, the book on analytic geometry by Demailly. –  Denis Serre Dec 7 '10 at 6:28
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Phil, at our present pace, the gap between parts II and III will be longer. –  Bill Johnson Jan 25 '11 at 8:10

Jeff Smith's book on combinatorial model categories.

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or his paper on ideals in ring spectra. –  Sean Tilson Dec 19 '10 at 4:12

Kervaire, Milnor: Groups of homotopy spheres II.

In the introduction to part I, they write:

"More detailed information about these groups will be given in Part II. For example, for $n = 1, 2, 3, \ldots, 18$, it will be shown that the order of the group $\theta_n$ is respectively:" (a table follows). Similar remarks are scattered throughout the text.

The details have been written down by other people and it must be said that part I contains the much more complicated arguments.

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In some sense, the sequel appeared; namely, Levine's paper "Lectures on groups of homotopy spheres" contains what he assumed would be the content of Kervaire-Milnor II. –  Andy Putman Mar 9 '11 at 19:47

The Igusa-Waldhausen paper (roughly) entitled,

The expansion space model for $Q(X_+)$

which is supposed to give a very different proof of the splitting $A(X) = Q(X_+) \times \text{Wh}^{\text{diff}}(X)$ that is based on a description of $Q(X_+)$ as the moduli space of finite relative cell complexes over $X$.

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At one point they were joking that each of them hesitated to put his name on the paper because he did not fully understand the other author's contribution, and that perhaps one way out of this impasse would be to publish it anonymously. –  Tom Goodwillie Mar 9 '11 at 12:26

This one is famous. It has been at the origin of a huge mathematical activity (conservation laws, homogenization, weak KAM, Hamilton-Jacobi equations, etc ...):

P.-L. Lions, G. Papanicolaou, SRS Varadhan. Homogenization of Hamilton-Jacobi equations

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one of my favorite unpublished paper :) –  Hung Tran Dec 7 '10 at 5:07

Here is a gap in a famous series of papers.

G.H. Hardy, and J.E Littlewood Some problems in Partitio Numerorum, VII

Their series of papers "Partitio Numerorum" is quite influential in the development of the Hardy-Littlewood circle method.

Some comments on the missing part are on page 253 in a paper by R.C. Vaughan, Hardy's legacy to number theory, Journal of the Australian Mathematical Society (Series A) (1998), 65: 238-266. Cambridge University Press

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4937088

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B. Farb. Automorphisms of $F_n$ which act trivially on homology.

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  • S. Gel'fand, Yu. Manin, Methods of homological algebra, first appeared in Russian as Методы гомологической алгебры. Введение в теорию когомологий и производные категории. Т. 1 (that is VOLUME 1). Volume 2 has been given up and the Springer Western edition does not cite Russian original, has many typing errors in formulas which Russian original does not have and it scraped off the tome 1 from the title.

  • M. Demazure, P. Gabriel, Groupes algebriques, tome 1, Mason and Cie, Paris 1970 -- later volumes never appeared

  • Z. Semadeni, Banach Spaces of Continuous Functions, Polish Scientific Publishers, Warzawa, 1971, never appeared from the Polish Sci. Publ. There is however a different book with a similar title in Springer in 1982, Schauder bases in Banach spaces of continuous functions. Lecture Notes in Mathematics 918. Springer 1982. v+136 pp. MR83g:46023.

  • John W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer-Verlag 1974. xii+282 pp. has been envisioned as a m3 volume project on formal category theory, some material is mentioned in volume 1 and never appeared. The monograph is very innovative and some of the material from the latter volumes was undoubtfully sketched by the author in some detail. The author later drifted to theoretical computer science.

  • John Duskin started a paper in several parts "Nerves of bicategories", part I appeared with great delay, partly due serious health problem the author experienced few years ago. Second and third part did not appear, although the contents description looks very promising. We wish the author good health and more to be seen!

Grothendieck planned not only later EGAs but also later SGA (e.g. some Berthelot's works in SGA 8). Bourbaki Elements are of course never finished as well (an now are very slow, asymptotically stalling) as the German encyclopedic work by Klein's students at the beginning of the 20th century. M. M. Postnikov wrote two volumes of a course on algebraic topology in Russian about basics of homotopy theory and promised the homology in "next semester" but no books appeared on that.

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Zoran, I am holding in my hand Banach Spaces of Continuous Functions, Volume 1, Warsawa 1971. Do you mean that it is Volume II which never appeared? –  Yemon Choi Mar 10 '11 at 0:17

Kurt Gödel referred to part II (Der wahre Grund für die Unvollständigkeit, welche allen formalen Systemen der Mathematik anhaftet, liegt, wie im II. Teil dieser Abhandlung gezeigt werden wird, darin, daß die Bildung immer höherer Typen sich ins Transfinite fortsetzen läßt) in his seminal paper Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik 38 (1931) p. 191. This part never appeared.

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There is a result by Oesterle, that proves that you can find the first non residue quadratic modulo a prime in no more than $70\log(p)^2$ step assuming the GRH, this result was then improved by Bach who replaced the constant $70$ by $2$. The result of Oesterle was never published and when I asked him why, he told me because the laptop containing the proof was stolen from his car. However I think he exposed his proof to the mathematical community, so it is widely recognized.

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A. Bertrand-Mathis, Le $\theta$-shift sans peine

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J. Berge. Some knots with surgeries yielding lens spaces.

(c. 1990; cited by 92 on Google Scholar.)

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W. Crawley-Boevey. The Deligne-Simpson problem.

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"The Aarhus integral of rational homology 3-spheres IV," by Bar-Natan, Garoufalidis, Rozansky and D. Thurston, never appeared. I think developments in the field overtook the need for the paper, which was referred to in the first paper in the series. This is a great series of papers by the way. Very clearly written.

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What were to be its results, I think, are the contents of a very recently completed preprint by Kuriya, Le, and Ohtsuki: front.math.ucdavis.edu/1005.3895 I have a different theory as to why "IV" never came out- that understanding the relationship of the Aarhus integral with the Ohtsuki series is a harder problem which requires more techniques, and the authors might not have fully appreciated that at the time "I" was written. I'm very happy that it's done now! –  Daniel Moskovich Dec 19 '10 at 0:49
    
I'm not sure what you mean by "developments in the field overtook the need for the paper"... to which developments were you refering? –  Daniel Moskovich Dec 19 '10 at 2:10
    
I actually was basing my theory on an overheard remark of Stavros Garoufalidis. He was asked if IV was ever coming out, and he said something like "Now that X has appeared, there doesn't seem to be as much need for this." However, I don't know what X was. –  Jim Conant Dec 19 '10 at 2:25

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