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It's a common observation in Lie theory that Cartan matrices and the Killing form are named after the wrong people; they were discovered by Killing and Cartan, respectively. I remember learning about many other examples of this phenomenon, but can't think of too many at the moment. Wikipedia has some examples here and here, but I'm curious about more obscure examples.

Bonus points for an interesting story behind why the concept was incorrectly named. Concepts that were deliberately named in honor of another mathematician don't count.

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closed as no longer relevant by Kevin H. Lin, unknown (google), Qiaochu Yuan, Zev Chonoles, Gjergji Zaimi Jul 9 '11 at 3:12

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By the way, in the mathematical community "Stiegler's law" is often referred to as "Arnol'd's law", inclusive of the corollary "Arnol'd's law applies to Arnol'd's law as well". – pasquale zito May 10 '10 at 20:24
To further complicate things, there is also Whitehead's law: "Everything of importance has been said before by someone who did not discover it." – bhwang May 10 '10 at 21:38
Oh gosh, I could not imagine that there are SO many wrong names. Perhaps some day there will be a big important Brandenburg theorem, of course just because another one has proven it. ;-) – Martin Brandenburg May 10 '10 at 23:09
Not that I have a problem with the question per se, but "the wrong people" is pretty ambiguous. The first person to study something might not be the most deserving -- often a crucial application or popularizations trumps the actual innovation. Nor is it necessarily the case that the intent of the naming was to honor the inventor -- frequently the naming is done for reasons of analogy ("Euler systems" come to mind). – Cam McLeman May 11 '10 at 1:10
Stigler's law is called Boyer's Law by H.C. Kennedy in "Who Discovered Boyer's Law?" (Amer. Math. Monthly vol. 79 1972, 66--67). It says that "Mathematical formulas and theorems are usually not named after their original discoverers." The label Boyer's law was chosen because Boyer gave many examples of this phenomenon in his book A History of Mathematics. – KConrad Sep 8 '10 at 17:12

56 Answers 56

Many of the examples mentioned go back to earlier centuries, when insulated national traditions and slow communications promoted mistaken labelling of results and concepts. A much more recent example from the 1950s involves the notion of Bruhat ordering on a general Coxeter group, motivated at first by the example of finite crystallographic reflection groups in Lie theory. The name seems to have been suggested by D.N. Verma in the late 1960s. For some reason the ordering itself fails to appear (even in the exercises) in Bourbaki's influential 1968 Chapters IV-VI dealing with Coxeter groups, root systems, Weyl groups and affine Weyl groups. Deodhar and others propagated the term "Bruhat ordering" in their papers, and as late as 1990 I routinely used this term in my book Reflection Groups and Coxeter Groups. But by then Borel, who had gotten more deeply involved in sorting out the history of Lie theory, objected that the ordering was not at all found in Bruhat's development of the Bruhat decomposition but had occurred for Weyl groups in Chevalley's treatment of the partial ordering of closures of Bruhat cells (Schubert varieties) in the flag variety.

As a result many of us now try in principle to start with something like Chevalley-Bruhat ordering (shortened to Bruhat ordering) or even Chevalley ordering. But this runs counter to a large body of literature including the 1979 Kazhdan-Lusztig paper.

Side remark: While Coxeter was the first to recognize the special presentation of a finite real reflection group that led to the term Coxeter group in Bourbaki, the general definition owes at least as much to people like Iwahori and Tits. Coxeter was interested in traditional (often intricate) combinatorial geometry and not in Lie theory or its generalizations. But short labels are easier to invent and tend to stick.

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I don't think that insulated national traditions can be limited to earlier centuries. Aren't there many examples of differing attribution in the third quarter of the 20th century during the Cold War? – Carl Mummert Aug 7 '10 at 15:00
@Carl: Certainly there have been artificial political barriers to communication and travel in more recent times, involving the Soviet Union, China, etc. Even so, journal communication has been better than in earlier centuries. Under Mao the Chinese universities often got pirated photocopies of western journals, for example, except during the Cultural Revolution. Fortunately it's becoming harder for regimes to stifle email and internet use, though they keep finding new ways to try. – Jim Humphreys Aug 7 '10 at 17:28
What about Verma modules, then? My own recollection is that Harish-Chandra already used them in order to construct simple highest weight modules with dominant integral highest weights. On the other hand, Dixmier made a remark that Bernstein-Gelfands-Verma modules would have been a more justified name, although less practical. – Victor Protsak Aug 7 '10 at 18:51
@Victor: "Verma modules" are typical of many concepts which should be named after multiple people, but the name is not really wrong (just incomplete). It's true that the modules played a technical role in the uniform existence proof for simple highest weight modules, but not until Verma's 1966 thesis were these modules studied seriously in their own right including the infinite dimensional ones. The subtle error in Verma's false multiplicity 1 claim perhaps helped stimulate the further work of BGG, Jantzen, Kazhdan-Lusztig. All of them soon made deep contributions to "Verma" modules. – Jim Humphreys Aug 7 '10 at 19:22
Jim: Since you have mentioned Iwahori and Tits in connection with Coxeter groups, I am reminded of historical injustice in naming the $q$-deformation of the group algebra of $W$ the $\textit{Hecke}$ algebra (also by Bourbaki?), whereas it first appeared in the work of Iwahori and Matsumoto. George Lusztig has tried to correct this, but it may be too late now. – Victor Protsak Aug 8 '10 at 6:06

Jones' Conjecture. Jones does not even exist; it's a Western pseudonym of Chuan-Min Lee.

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Perhaps counterexamples to Stigler's/Arnol'ds law are actually the rare items. The most significant one that I know is the Cartesian coordinate system which, strangely, seems to have actually been invented by Descartes!

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Please note that Cartesius is the latin name under which Descartes was known. – Sebastian Goette Oct 2 '15 at 9:34

Farey series, attributed to Farey, were actually first studied by Haros.

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Nakayama's lemma was first proved by Krull in a special case, and by Goro Azumaya in the general case.

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Did Nakayama at least "discover" the lemma? I think this is also important because I think no one complains about naming Fermat's last Theorem not after Wiles. – Martin Brandenburg May 11 '10 at 11:18
I think that's why many people prefer to call it NAK lemma. – VA. May 12 '10 at 2:51
It's not so clear. In the historical notes to his book, Nagata says that in the case N=0 and M an ideal, the lemma was given by Krull, and that the general case is in a paper of Azumaya. But Nagata learned it from Nakayama and Azumaya when he was an undergraduate. Since some mathematicians were calling it Nakayama's lemma, he asked Nakayama who had this formulation first, to which Nakayama responded that he did not remember whether Nakayama or Azumaya was the first. Probably it should be considered folklore, and the name "Nakayama's lemma" seems appropriate. – JS Milne May 12 '10 at 11:34
"History is a myth men agree to believe."-Napoleon Bonaparte – The Mathemagician May 13 '10 at 14:32

Pell's equation

By a mistake of Euler, the Diophantine equation y^2 - Ax^2 = 1 has been erroneously known as "Pell's equation"; but, in fact, the English mathematician John Pell (1611-1685) did no more than copy it down in his papers, from Fermat's letters of 1657 and 1658.

For an extensive historical account on "Pell's equation", see Sir Thomas L. Heath, Diophantus of Alexandria : A Study in the History of Greek Algebra (Dover Pub., New York, 1931-1963, 552 pages), Supplement, Section II: "Equation y^2 - Ax^2 = 1, pp. 277-292. Particularly in page 285, after a presentation of the history of the equation up to Fermat's time (including citations to Pythagoreans, Archimedes, Diaphanous, and the Indian solution), one can read that:

" ... Fermat rediscovered the problem and was the first to assert that the equation x^2 - Ay^2 = 1, where A is any integer not a square, always has an unlimited number of solutions in integers. His statement was made in a letter to Frénicle of February, 1657 (cf. Oeuvres de Fermat, II, pp.333-4). Fermat asks Frénicle for a general rule for finding, when any number not a square is given, squares which, when they are respectively multiplied by the given number and unity is added to the product, give squares. If, says Fermat, Frénicle cannot give a general rule, will he give the smallest value of y which will satisfy the equations 61y^2 + 1 = x^2 and 109y^2 + 1 = x^2 ? (Footnote 3: Fermat evidently chose these cases for their difficulty; the smallest values satisfying the first equation are y=226153980, x=1766319049, and the smallest values satisfying the second are y=15140424455100, x=158070671986249)." And, after a extensive quotation of Fermat's letter, in page 286, one can read that: "The challenge was taken up in England by William, Viscount Brouncker, first President of the Royal Society, and Wallis (Footnote 1: An excellent summary of the whole story is given in Wertheim's paper "Pierre Fermat's Streit mit John Wallis" in Abhandlungen zur Gesch. der Math., IX. Heft (Cantor-Festschrit), 1899, pp.557-576). See also H. Konen, Geschichte der Gleichung t^2-Du^2=1, Leipzig (S. Hirzel), 1901). At first, owing apparently to some misunderstanding, they thought that only rational, and not necessarily integral solutions were wanted, and found of course no difficulty in solving this easy problem. Fermat was, naturally, not satisfied with this solution, and Brouncker, attacking the problem again, finally succeeded in solving it. The method is set out in letters of Wallis (Footnote 2: Oeuvres de Fermat, III, pp.457-480, 490-503) of 17th December, 1657, and 30th January, 1658, and in chapter XCVIII of Wallis' Algebra; Euler also explains it fully in his Algebra (Footnote 3: Part II, chap. VII), wrongly attributing it to Pell (Footnote 4: This was the origin of the erroneous description of our equation as the "Pellian" equation. Hankel (in Zur Geschichte der Math. im Alterthum und Mittlelalter, p.203) supposed that the equation was so called because the solution was reproduced by Pell in an English translation (1668) by Thomas Brancker of Rahn's Algebra; but this is a misapprehension, as the so-called "Pellian" equation is not so much as mentioned in Pell's additions (Wertheim in Bibliotheca Mathematica, III, 1902, pp.124-6); Konen, pp.33-4 note). The attribution of the solution to Pell as a pure mistake of Euler's, probably due to a cursory reading by him of the second volume of Wallis' Opera where the solution of the equation ax^2 + 1 = y^2 is given as well as information as to Pell's work in indeterminate analysis. But Pell is not mentioned in connexion with the equation at all (Eneström in Bibliotheca Mathematica, III, 1902, p.206)."

For more information about "Pell's equation", see Harold M. Edwards, The Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Springer-Verlag, New York, 1977, 410 pages), pp. 25-33. Particularly in page 33 one can read that

"This problem of Fermat is now known as "Pell's equation" as a result of a mistake on the part of Euler. In some way, perhaps from a confused recollection of Wallis's Algebra, Euler gained the mistaken impression that Wallis attributed the method of solving the problem not to Brouncker but to Pell, a contemporary of Wallis who is frequently mentioned in Wallis's works but who appears to have had nothing to do with the solution of Fermat's problem. Euler mentions this mistaken impression as early as 1730, when he was only 23 years old, and it is included in his definitive Introduction to Algebra written around 1770. Euler was the most widely read mathematical writer of his time, and the method from that time on has been associated with the name of Pell and the problem that it solved --- that of finding all integer solutions of y^2 - Ax^2 = 1 when A is a given number not a square --- has been known ever since as "Pell's equation", despite the fact that it was Fermat who first indicated the importance of the problem and despite the fact that Pell had nothing whatever to do with it."

See also André Weil, Number Theory : An approach through history - From Hammurapi to Legendre (Birkhäuser, Boston, 1984, xv+375 pages), in many different pages. In particular, at page 174, one can read that:

"Pell's name occurs frequently in Wallis's Algebra, but never in connection with the equation x^2 - Ny^2 = 1 to which his name, because of Euler's mistaken attribution, has remained attached; since its traditional designation as "Pell's equation" is unambiguous and convenient, we will go on using it, even though it is historically wrong."

Raul Nunes ( ) NEST Nunes' Exposition of Scientific Truths ( )

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The algebraic numbers that are now commonly called "Gauss sums" were studied in more general form than that introduced in Gauss's Disquisitiones by Lagrange [1]. In that same work, Lagrange shows how to generate an abelian extension of degree n by adjoining an nth root after, if necessary, adjoining the nth roots of unity. These generators were later called "Kummer generators". Jacobi sums, which are closely related to Gauss sums, were studied earlier than Jacobi by Gauss and Cauchy.

Finally, a story best recounted by Weil [2]: "For reference, we recall that the Gauss sums appear among the local constant factors in the functional equations of the $L$ functions; these factors are also called "nombres radiciels" ("root-numbers", "Wurzelzahlen"), undoubtedly because of Hilbert, who a had a sort of genius for bad terminology, where it would have been sensible to name "Wurzelzahl" that which before him had been named a "Lagrange resolvent" , and "Lagrangian Wurzelzahl" that which here has been called a Gauss sum".

[1] Lagrange, Reflexions sur la resolution algebrique des equations, Nouveaux Mem. de l'Acad. R. des Sc. et B.-L. de Berlin, 1770-1771 = Oeuvres, vol. III, p. 332;

[2] Weil, La Cyclotomie Jadis et Naguere.

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If the Russians are to believed, Kuratowski's theorem in graph theory was proved earlier by Pontryagin, but he hadn't published his notes.

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In reference to exactly this phenomenon (and in particular to the case of Pell's equation), Andre Weil once observed that "This has happened many times in mathematics. For example, I live on von Neumann Circle. I live there. Yet still it is called von Neumann Circle".

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The Shimura-Taniyama conjecture was originally known as the Weil conjecture see, also see the comment of Weil on page 7 (with other examples) in his response to Lang on the same issue as in the question posed here.

Additionally, the Frey curve was actually first considered by Yves Hellegouarch.

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I think there is also some commentary on this situation in Shimura's The Map of my Life, which is quite an interesting read either way. – Qiaochu Yuan Jan 19 '11 at 16:59

Pythagoras' Theorem apparently predates Pythagoras.

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And similarly much of Euclid predates Euclid. – Tom Smith May 10 '10 at 20:51
But so far as we know, Pythagoras (or one of his followers) was the first to prove it, or even to attempt to do so. – Charles Staats May 11 '10 at 2:39

Euler's nine point circle was never discussed by Euler. This is an error of the "argument by authority" type: Catalan propagated that incorrect attribution made by another scholar, the "learned Terquem", without checking it himself.

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The "Lichnerowicz formula" relating the square of the Dirac operator to the Laplacian has been proved more than 30 years earlier by Schrödinger.

See: E. Schrödinger, Dirac'sches Elektron im Schwerefeld, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1932, 105-128 (1932).

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Farey fractions were introduced by C. Haros. John Farey asked a question about them that reached Cauchy, and Caucy then attributed the question and result to Farey, and the rest is history.

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It seems to me that Lagrange's theorem may well be one of the most prominent examples of the phenomenon under discussion.

According to J. J. Rotman,

the theorem was inspired by work of Lagrange (1770), but it was probably first proved by Galois.

Curiously enough, the Wikipedia article adscribes the first complete proof of the theorem to Pietro Abbati Marescotti.

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Some people have tried to give examples with as high a multiplicity as possible. I want to try to break the record for the smallest non-zero example: Martin's axiom was introduced by Martin and Solovay. (I judge that to have multiplicity 1/2.)

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Sorry, but this is a non-example. Martin's axiom was introduced by Martin. It is true that this happened in a paper by Martin and Solovay, and that it was after a construction of Solovay and Tennenbaum suggested it, but there is no misnomer here. – Andrés E. Caicedo Oct 13 '10 at 20:58
So much for trying to find examples on Wikipedia. But perhaps that actually improves my example, making its multiplicity even smaller than 1/2. – gowers Oct 13 '10 at 21:36

I had a companion observation that almost noone attributes the well-known sum-of-roots, product-of-roots etc. polynomial formulas as Vieta's formulas as I posted on Yahoo!Answers.

Because as user absird pointed out, it makes that sort of topic Google-proof; at least a bad name is better than no name for purposes of searching or discussion.

('Yes it's very hard to refer to something when noone knows it by its proper name or uses that name. I tried many Google searches on "sum-of-roots product-of-roots" and it was almost impossible to get a coherent lead.')

MathWorld notes: The theorem was proved by Viète (also known as Vieta, 1579) for positive roots only, and the general theorem was proved by Girard.

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Really? People in high school competition circles, at least in the US, are generally pretty good about calling them Vieta's formulas. – Qiaochu Yuan Jan 18 '11 at 20:10
It may be geographically dependent. At high school level, no. I was educated in Ireland and never saw these referred to as Vieta's formulas. Even in college they would be blankly presented as 'properties of polynomials'. If you google (and manage to find them as 'sum-of-roots') that is typically how you will find them presented. At IMO level, perhaps. Even then I don't recall ever seeing the name (not like Cauchy-Schwarz, Hölder, Pappus, Chinese Remainder Theorem et al). Personally I only found out the name in 2008, i.e. 20 years later. And it's not like I didn't read a lot. – smci Jan 18 '11 at 20:38
I was educated in the US, and was unaware of the name "Vieta's formulas" until about two minutes ago. – Charles Rezk Jan 18 '11 at 20:48
@Charles: Uhuh. What name, or topic, were they taught to you under? ('properties of polynomials'?) – smci Jan 18 '11 at 21:12

A favorite of mine is l'Hôpital's rule. l'Hôpital paid Johann Bernoulli a retainer to keep him updated on developments in calculus and to solve problems he had. Correspondence shows that Bernoulli stated and proved the rule, which l'Hôpital then published.

Heine-Borel was first published by Borel, not Heine. In fact, Heine's name was attached because he was using similar methods to solve related problems. Too bad for both of them that it was actually Dirichlet who was the first recorded to have proved it, but his notes were published posthumously and after Borel's proof.

Cramer's Rule was published first by MacLaurin, and some believe MacLaurin knew the proof some 20 years before Cramer's publication.

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On l'Hôpital, see here. – Andrés E. Caicedo Nov 7 '13 at 22:20

Morse theory is named after Marston Morse; it was widely used at least 50 years earlier. Wikipedia mentions Cayley and Maxwell, in the context of topography. Maxwell also used it in his work on electromagnetism, as detailed (complete with extensive passages from Maxwell's treatise) in the appendix of Mystery of point charges (A. Gabrielov, D. Novikov and B. Shapiro) available here (subscription)

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I'm not sure this is a fair example.The generalized theory of critical points in Euclidean space was used to be sure far earlier then Morse-but Morse and his contemporaries made the vital step of generalization to the topological language of manifolds. I'm not sure we should call what Maxwell and the others used strictly speaking Morse theory,but it's direct ancestor in Euclidean space. – The Mathemagician Oct 13 '10 at 21:03
@Andrew L: At the same time, it appears unfair to penalize Maxwell when the topological setting in question was not available at the time. Maxwell probably did all he could do within the limited framework of his era, and while I readily acknowledge and appreciate Morse's contribution, Maxwell hardly gets any credit at all for what he did achieve. – Thierry Zell Oct 13 '10 at 22:01
@Thierry Maxwell and the others certainly deserve credit for making critical progress in Morse Theory's development,but we clearly can't call what they were doing Morse theory by modern standards,any more then we can call the work of Issac Barrow calculus since he didn't recognize the connection between the quadrature (integration) problem and the tangent (derivative) problem. Of course,that does nothing to invalidate the historical importance of Barrow's work with limits,which was the stepping stone to Newton and Lebnitz's breakthroughs. – The Mathemagician Oct 14 '10 at 10:27

Liouville talked about the Legendre function when he studied the so-called Euler Gamma function. It made me doubt about who defined the Gamma function first.

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According to this link Steiner Systems were mentioned by by W Woolhouse in 1844 before the famous Kirkman Schoolgirl problem (P Kirkman 1847) - Steiner's work was more systematic and did advance the theory, but it came in 1853.

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I don't know if this is a real example, but it led to a nice gem in a recent abstract on the arXiv: "Glaisher's correspondence goes back to Euler."

(As far as I know Glaisher generalized Euler's bijection, which is why he gets the eponym -- in addition some people say "Euler-Glaisher" instead.)

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Huh? I don't follow. Euler did not have "bijective proof" as a concept. It was Glaisher who invented the bijection, but if I recall correctly never published it - Sylvester did it and attributed it to him some time later. You are probably referring to Euler's "odd vs. distinct" partitions theorem. Using modern "involution principle" technology one can convert analytic proofs into bijective, and in this case the one-line Euler's proof becomes Glaisher's bijection (this is O'Hara's theorem - see my survey on partitions and a recent paper on O'Hara algorithm, joint with Konvalinka). – Igor Pak May 10 '10 at 21:50

Cartan discovered the Killing form, and Killing discovered the Cartan matrix.

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That's the example I used in the original post :) – Qiaochu Yuan May 13 '10 at 1:44
Aw, nuts, that's what I get for not reading things carefully. :) – David Hansen May 13 '10 at 2:08

I think the Kazhdan-Lusztig Conjectures are due to Vogan.


True or false, the claim is mainly based on the very first two paragraphs of

[II] Irreducible characters of semisimple Lie groups II. The Kazhdan-Lusztig conjectures. David A. Vogan, Jr. Duke Math. J. Volume 46, Number 4 (1979), 805-859. --- The link

gives a universal access to the first page, which contains the two paragraphs in question. In case you don't have access to the full paper, here is a scan of the references (to completely understand the two paragraphs):

Here are two more references:

[I] Irreducible characters of semisimple Lie groups I, David A. Vogan, Jr., Duke Math. J. Volume 46, Number 1 (1979), 61-108.

[KL] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Inventiones Mathematicae, Volume 53, Number 2, 165-184.

I would summarize things as follows.

Step 1. In [I] Vogan made a certain conjecture.

Step 2. [II] and [KL] were written simultaneously. Each paper cites the other. In [KL] Kazhdan and Lusztig also made a certain conjecture. When he learned this, Vogan immediately (or at least very fast) proved that the "Step 1 conjecture" implies that of Kazhdan and Lusztig. (He even showed that the "Step 1 conjecture" generalizes that of Kazhdan and Lusztig.)

But, again, the best is to read carefully the first two paragraphs of [II]. Vogan explains this much more clearly than I, and it's always better to hear things from the horse's mouth.

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I never heard that! (Although the names of Goresky and MacPherson are sometimes mentioned). What is the source? – Victor Protsak Jun 28 '10 at 15:24
The answer is incorrect, since the KL 1979 paper (Invent. Math.) was the first to state the KL Conjecture based on Hecke algebras and KL polynomials. Kazhdan and Lusztig were motivated by questions about singularities of Schubert varieties, Springer's representations of Weyl groups, Jantzen's work on Verma modules, etc. Vogan was approaching similar machinery from the direction of real Lie group representations, which led him after the KL paper to complete parts of his own program using "KLV polynomials". See Steven Kleiman's history of intersection homology (arXiv) for the role of GM. – Jim Humphreys Aug 7 '10 at 16:00
Dear Victor, I've just discovered your comment! Usually I get an automated alert whenever somebody comments in one of my answers. I could almost swear I didn't get any for your comment. Next time, don't hesitate to email me! (My address is very easy to find, for instance on my MO page.) – Pierre-Yves Gaillard Aug 7 '10 at 18:10
Dear Victor, dear Jim: Thank you very much for your comments. I'll try to explain how I view things in an edit (soon to come) to my answer, and you'll tell me why I'm wrong. In a nutshell, I'm referring to what Vogan did before, and not after, the Inventiones KL paper. Things are very clearly explained in the very first two paragraphs of… Also I made this claim several times in presence of David Vogan, and he never objected. (I'm sure you know his honesty and modesty.) – Pierre-Yves Gaillard Aug 7 '10 at 19:49
I think this is mostly a labelling question, since what most people have referred to over the years as the Kazhdan-Lusztig Conjecture (soon proved geometrically by Beilinson-Bernstein, Brylinski-Kashiwara) concerns multiplicities in Verma modules. What Vogan developed was in a sense "equivalent" but not the same statement in spite of his modest label. The KL formulation has had natural analogues in other kinds of representation theory. And Jantzen's older conjecture on Verma module filtrations was later seen to imply it, but the formulations are quite different. – Jim Humphreys Aug 7 '10 at 21:10

The notion of Frobenius manifold is due to Dubrovin

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Why -2 votes??? – Yougeeaw Aug 31 '11 at 10:17

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