Professor Aczel raises an issue of
received wisdom: That it's not the
referee's responsibility to vouch for the accuracy of an author's
paper. He quotes R. Boas as Chief Editor on the Monthly of the AMS. He
suggests I didn't make a case for this responsibility though I did
at least 50% of the papers I referee have serious errors.
I heard this received wisdom first from M. Artin when I was a semester
at MIT as a Sloan Fellow. My response – not agreeing with Artin, with
whom I often conversed – was that the seriousness of errors depends on
the article's ambitions and use of sophisticated mathematics. Also, if
you don't fine-tune this opinion for mathematics, where would you?
Finding referees for new, not-yet-assimilated tools is one tough
problem. It requires interdisciplinary refereeing expertise and editors
who follow the analysis. Or else, it is rife for abuse, applying stingy
analysis to one author not applied to another.
An example will help convey what I mean by a serious error. A recent
submission to Math. Annalen aimed to produce infinitely many primes
supporting a conjecture of Ihara from the early '80s. Amidst technical
details the author used the Galois closure of curve covers of
prime-power degree as if the result would be a p-group. (Consider
prime-power degree symmetric groups to see how wrong this is.) Only the
2 of his argument could be salvaged. Though then, the paper was
inappropriate for the announced expectations of this journal. Yet, the
initial editor accepted it and the author thanked the editor for
catching the error.
Refereeing and proofs in practice:
Two examples from my early experience informed my response to
They also help me address the full gamut of Aczel's concerns. My first
paper solved Schur's 1923 Conjecture [S]. There
was a 500+ paper
bibliography sent to me of earlier papers considering this
easy-to-state conjecture. Most on particular polynomials not violating
it. My short paper characterized properties an example would have.
Then, using Riemann surfaces and group techniques, uncommon to the
to show only the list of Schur could so contribute. The proof was
My first Annals paper [AK] solved the main
general problem posed by Ax and
Kochen coming from their proof of a famous E. Artin conjecture. Prior
to my result, someone well-connected to the area proved there could be no such
theorem as mine. My paper had five referees. Four called my office at
Stony Brook, with the same technical questions on my generalization of
Chebotarev's theorem. A 5th revealed himself years later.
I got a fair hearing from a rare fair editor – Armand Borel – to whom I
lobbied for my paper during a two year post-doctoral at IAS. My
experience: Editors dominate in this process over referees.
Recieved Wisdom: Here
are other points of received wisdom complicated by the various
parties and vested interests in the production of journals.
Other considerations often annihilate the grains of their truth. The
final paragraph of Aczel's letter gripes about referees: that many
won't respond to editors after receiving a refereeing request.
- Good work will get recognized [in the end].
- It's the judgment of your peers that counts.
- Most editors (resp. referees) are fair/work hard [see Bryan
Cain's August 2007 Notices lament on
the failure of this].
- It's the author's job to communicate well [immediately, prior to
and independent of any referee feedback].
- Anyone can referee. What matters is results.
Checking the rewards for
refereeing: Yet, this was half the target of Should
Journals compensate Referees?. To pay/reward talented referees
(responses to my article included further suggestions) would develop a
responsive referee cadre. Still, how about those editors who take lack
of referee response as if it is the author's fault?
Correspondent #5 in my article
asked, Why do mathematicians referee
without compensation? He wondered, among the possibilities if
it was a fair trade for having
their papers refereed?
The experience for my last paper [L] sent out for refereeing says
no, you might not even get that. [AK] and [S] from the
beginning of my career were (respectively) key references for two
papers – [A] (single-authored) and [B]
(3-authored) – an IMRN editor asked me to referee.
Expansion from [AK] governed
the first half of [A]. Vovodsky's relative Chow Motives – initially a
black box to me – governed its second half. I opened direct
communication with the paper's author for further clarification, but
the editor said I was holding up the process and accepted the paper. A
year later the author substantially revised and resubmitted his paper
because of an error in the area of my questions.
result of [D] published in 2005
was the main result of [B] on
which two of [B]'s authors had seen me lecture. I accepted the paper
for publication with considerable comment separating what they did from
previous results, but also asked they at least
change their abstract. This consisted of just a special case for curves
of [D]'s topic-opener. The authors took many months to gave a
response, refusing to acknowledge [D]. The IMRN editor accepted
their paper without further revision, saying despite my immediate
referee report, that the process had gone on long enough.
months here is the (same) editor's response to my submission [L]. "We have not
been able to find a referee
for your paper
IMRN/82303 entitled 'Alternating Groups and Moduli Space Lifting
Invariants.' In fact we get no answers whatsoever. We've decided not to
continue but to simply return the paper without comment for submission
What is at stake?: There are
many players in this process, including publishers who add little to
our TeX-ed papers, and editors who edit precious little. The more
players, the easier for any one to get off the hook. My opinion piece
said: Refereeing is a hard task, and too few do it well. With more
incentive, more mathematicians might develop the
high skills that go with quick, quality refereeing.
I add here: Likely, too, editors would help improve the refereeing
process by informing researchers more precisely of their journals'
goals and analytical standards. Claiming, as many journals do, high
standards, is not much guide for whether a particular paper belongs in
a given journal.
I think it significant that another article in the May 2007 Notices
was on the issue of journals. This was the announced resignation of
Topology Journal editorial
board. The disparate issues, however, raised
by this and my opinion piece don't reconcile easily. The former appears
as a revolt of elite mathematicians against
venal publishers, while the latter insists on rectifying internal
(to mathematics) difficulties between referees and editors. Further, as
stated on p. 906 of [AMS07], 78% of the AMS
operating budget is for publishing, especially aimed at keeping
publication costs down.
comments on those issues by reconsidering some aspects of the Received
Finally, my opinion piece started with the plight of a
talented (I claim) mathematician with difficult
dealings with the researchers in his area. So, it is hard to
find referees for his ambitious projects. He's an example of
another unanalyzed topic: Which mathematicians have the most troubles
with getting timely refereeing? I contend some with the most
difficulties are ambitious,
technically and imaginatively, and the more modest and less
reaching might find they get in print more
easily. [IM] includes
comments from some who wrote directly to me after my opinion
piece. This amounts to a thesis, likely testable in several
ways. Yes, qualifiers relate to the authors' having support
within well-recognized schools.
[IM] makes two points on the ethics of
mathematics: more on the received
wisdom list above, and noting that the essence of mathematics –
it's ability to define precisely, and then prove things from these
definitions – fails to promulgate proof to the rest of science. My
suggested reason – over specialization in the mathematics community
itself – further
motivates encouraging quality referees to rise above their
[A] J. Nicaise, Relative Motives
and the Theory of Pseudo-finite fields, IMRN to appear.
[Ac] J. Aczel, Response
to "Should Journals pay Referees," October
2007 Notices AMS.
[AMS07] From the AMS Secretary, Notices of the
AMS Aug. 07 54, #7, 904–915.
R. M. Guralnick, T. J. Tucker and M. E. Zieve, Exceptional Covers and Bijections on
Rational Points, IMRN to appear.
[AK] with G. Sacerdote, Solving
diophantine problems over all residue class fields of a number field
..., Annals Math. 104
(1976), 203–233. Picked up
(with Ken Ribet's help) from JStor
Introduces the Galois Stratification procedure in its Original,
geometric form. Material corresponds roughly to Chap. 25 of the Fried Jarden book
[D] The place of exceptional covers among all
diophantine relations, J. Finite Fields 11 (2005) 367–433.
[IM] Who Reads Your Papers?: Achievement vs Success, Part I?
Andy Magid is departing and Steven Krantz is entering editor of the
Notices. Magid's article covered territory from the large AMS Notices
changes starting in 1995. I use the events that precipitated those
changes, at AMS and in departments, to discuss parallel changes
in refereeing/editing. It suggests there was another referee paradigm
that is still relevant today, but little used. Part II, in preparation
12/03/09, suggests it won't work to plead with everyone to be a
better referee. Rather, it suggests incentives journals could
apply to encourage those with the appropriate skills and interest to be
the referees we really need.
[L] Alternating groups and moduli space
lifting Invariants, Israel J. Math. (2009), 1–68, Arxiv #0611591v4. Starts with a description and properties
of spaces of 3-cycle covers, Arxiv #0611591. Serre asked me a question
with an incorrect guess at the answer. I gave a method that hinted at
the right answer, proved it in special cases and suggested a result
with further application. Serre agreed, used part of the method, and
completely answered his version of the question [Se].
I waited 10 years to write out three applications – two at the end of
the paper justifying my version of the result, one to producing special
canonical theta nulls on Hurwitz spaces, the other to analyzing a
possibility for a presentation of GQ (a la, the proven, but less
canonical presentation in [FrV2]).
[S] On a
conjecture of Schur, Mich. Math. J. 17 (1970), 41--45.
[Se] J.-P. Serre, Revêtements
a ramification impaire et thêta-caractèristiques,
C.R.Acad. Sci. Paris 311