From Wikipedia:

“The Séminaire Nicolas Bourbaki (Bourbaki Seminar) is a series of seminars (in fact public lectures with printed notes distributed) that has been held in Paris since 1948. It is one of the major institutions of contemporary mathematics, and a barometer of mathematical achievement, fashion, and reputation. It is named after Nicolas Bourbaki, a group of French and other mathematicians of variable membership.”

As you can see from the link, a Bourbaki talk is always given by a speaker about other people’s work. It is accompanied by an expository paper explaining difficult recent material.

It is an honor and a pleasure of me to give a Bourbaki talk on Babai’s major breakthrough. This has been an exciting story.

I understand that my talk will be streamed live: live video. The notes will be made available online in the very near future. I hope they will make it easier for all others to follow the proof themselves.

Preparing a Bourbaki talk implies (a) going through somebody else’s work in great detail, and (b) preparing an expository paper on the subject. My exposition of Babai’s work is ready and will be publicly available in the next few days. It is a complete walkthrough of the proof, and should allow others to verify, as I did, that the modified version is correct.

Here is an excerpt from my introduction (in French in the original):

“Thm 1.1 (Babai) .- The string isomorphism problem can be solved in time exp(exp(O(sqrt(log n log log n)))) for strings of length n.

It is clear that the bound here is sub-exponential, but not quasipolynomial. In November 2015, Babai announced a solution in quasipolynomial time, with an explicit algorithm. The process of preparing this expository work confirmed that the algorithm was correct, or easily repairable, but it also made me realize that the time analysis was incorrect. The version announced here is correct.

Thm. 1.2 (Babai).- The graph isomorphism problem can be solved in time exp(exp(O(sqrt(log n log log n)))), where n is the number of vertices.

Our main references will be [Babai’s 2015 preprint and his extended abstract at STOC 2016]. We will attempt to examine the proof in as much detail as an expository work of this format allows, in part to help eliminate any doubt that may remain on the current form of the result. The error lay in a part of the proof that can be isolated and corrected, and might be later improved on its own (“Split or Johnson”). The rest of the work – rich in innovative ideas – is still valid.”

See also Laci Babai’s own announcement on the subject:

https://people.cs.uchicago.edu/~laci/update.html

Update (Jan 14): As many of you know, Babai posted last Monday that he had fixed the problem. This is just to tell you that (a) his fix is correct (and elegant), (b) I have spent the last five days checking other things in the proof and making some bits explicit. I can now state with assurance that the proof is correct: Babai is now giving an algorithm that works in quasipolynomial time. See the video of my Bourbaki talk at https://www.youtube.com/watch?v=7NR975OM2G8&list=PL9kd4mpdvWcCN64K5VhaYFH_gBa-WlIr3&index=1

I will put up the corresponding expository article soon.

Summer was full of activities and travel. After giving a course at the summer school on analytic number theory at IHES, I went to Korea for most of August (ANTS and of course the ICM). Then I went back to Paris and got the keys to my new office at IMJ (Paris VI/VII), where I will be from now on as a CNRS Directeur de recherche. The office is in the Paris VII campus, close to the Bibliothèque Nationale and, most importantly, the Cinémathèque.

… and then I left for Saint Petersburg, to spend a trimester there as a Lamé Chair. During that time, I was offered a Humboldt Professorship (at Göttingen) , which I am now considering.

I have been back in Paris since late December, after relatively brief trips to Perú and the Czech Republic. I also managed to lose the keys to my office in the interval. It took a very long time to get another set of keys.

On a different note – my survey paper on growth in groups got accepted; it is about to appear. I also spent a great deal of time rewriting my proof of the Ternary Goldbach Conjecture – it is now in an essentially self-contained monograph. Besides adding expository material, I simplified some parts of the proof – notably the part on parabolic cylinder functions.

I have some partial drafts of planned blog posts from the last few months. Let me include here a brief set of impressions from my Korea trip. I did this in French, just to keep my chops up.

Ma première conférence en Corée a eu lieu – comme il est habituel dans certains pays – dans un hôtel, quelque peu isolé du reste de la ville de Gyeongju. En route de la gare, il était possible de voir un autre visage de la Corée que celui d’un pays hautement industrialisé; au fait, des parties de Gyeongju ressemblent à certaines villes de province d’un pays en développement, avec une intense activité commerciale conduite dans des petits locaux d’aspect plutôt pauvre. L’hôtel, au bord d’un lac, était luxueux, au moins dans ce que concernait ses espaces communs; il portait le nom d’un des conglomérats qui dominent la vie économique du pays. Après peut-être une demi-heure, j’ai eu envie de partir et voir un peu du monde hors ses murs. Pour être précis: je suis parti avec l’intention de trouver du 청국장 – c’est-à-dire, le fameux tofu puant coréen.

Je disposait de peut-être dix mots de coréen. Heureusement, le chauffeur du taxi commandé par l’hôtel n’était pas seulement un homme de bonne volonté, mais aussi très expressif, au point que j’arrivais à croire que je comprénnais une petite partie de ce qu’il me disait. Il m’a laissé à l’entrée d’un petit restaurant traditionnel qui n’avait pas du tofu puant. Là, quelqu’un m’a dirigé vers ce qui s’est avéré être un atelier semi-urbain où, à juger par un odeur très prometteur, du tofu puant était fabriqué. Malheureusement, l’atelier était fermé.

J’ai retrouvé le taxi, lequel reprenait sa route vers la ville. Après assez d’efforts, le chauffeur a trouvé un restaurant populaire pas loin de l’hôtel; après avoir vérifié qu’au moins un plat à base de tofu puant était servi, il est parti en refusant d’accepter qu’une partie de ce que le taximètre indiquait. En très peu de temps, je me suis trouvé face à un festin destiné qu’à moi, consistant en un ensemble de petits plats, y inclus du poisson salé, des grandes feuilles ressemblant à la menthe (mais en beaucoup mieux), et, bien sûr, du 청국장.

Ce dernier ne puait pas du tout; il avait plutôt un bouquet qui ressemblait à ceux de certain fromages, a savoir ceux dont on dit qu’ils puent.

Something that truly excited me about Korea was the possibility to use my newly-found knowlege of the Korean script. Indeed, it really helped me to get around, even without any actual knowledge of Korean. I had studied the Korean script twice: once, when Don Zagier taught me the basic principles in twenty minutes after dinner at the British Mathematical Colloquium; the second time, when I skimmed the wikipedia page on it on the morning of the day when I flew to Korea.

As you might have gathered, it is, so to speak, very logical – in fact, it was designed specifically for the Korean language, by a small committee of rather talented 15th-century people. The name it was given once by some (아침글, “one-morning writing”) was presumably meant to be pejorative, but it does reflect how quickly one can learn it. Or, to put it in a much more complex script, 故智者不終朝而會，愚者可浹旬而學 (or so I read).

A mystery remains: why is it that, in Korea, good local food is cheap, as are alcoholic drinks, but coffee is expensive? Or is this just the price that people pay to be seen in a Westernized coffee shop – that is, can one get less expensive coffee elsewhere?

Let me first announce the following conference, which will take place in St Petersburg at the end of November:

I hope it will turn out rather well. Please tell us if you want to attend – though, as we have limited funds, you would essentially have to fund your trip through your own grant (or your advisor’s, if you are a student, but we might be able to help with lodging in that case.) Things may be easier if you are in Russia. All are welcome!

Besides offering some tables, the article briefly enumerates a few perspectives on the movement of mathematicians between countries, without really endeavouring to resolve the tensions between them. My aim here will be to give a summary of the situation and what I see as a second step in the analysis of these issues, in part with the hope that readers of this blog will discuss them further.

In case you have not looked at the tables yet: the three countries in which the most speakers were born are France (30 speakers), the Soviet Union (27) and the United States (26); the next countries in the list, far behind, are Germany and the UK (12 each). This should be completely unsurprising to anybody within the mathematical community. Then we have Italy and China, each at 9, and Hungary at 8. (South America – indeed all of Latin America – has a grand total of 7, including 3 from Argentina and 2 from Brazil.)

The birth-to-PhD and PhD-to-work charts show that the US acts as a very strong attractor for prospective doctoral students (58.5 speakers born elsewhere got their doctoral degrees in the US, and no speakers born in the US got their PhDs elsewhere; here, fractional numbers indicate shared positions/studentships and the like), but not as a workplace (28.5 speakers left after getting their PhD in the States, and 21.5 moved to the States after getting their PhD elsewhere). France works almost as a closed system at the educational level (only 3 speakers born in France got their PhDs elsewhere, and 3.5 did the inverse move) and as a very mild attractor as far as jobs are concerned (8.5 foreign PhDs moved to France, including 4 from the US, and 4 French PhDs moved out of France). As for the Soviet Union – 23.5 out of 27 speakers born there live outside the successor states (13 of them in the US), 10 moved out already to get their PhDs (6 of them in the US), and nobody not born in the Soviet Union currently works in the successor states. Only 3 of the 16 speakers born in Eastern-Europe-minus-USSR got their PhD there, and only one of them works in Eastern Europe now (a Hungarian in Hungary). Again, the overall picture agrees roughly with what conventional wisdom would have expected.

As for Latin America – 7 speakers were born there, and 3.5 left for the US to their PhDs, with the others staying in their home countries (Brazil and Argentina); 2 work in the US, four work in their home countries (again, Brazil and Argentina) and one works in France (myself). Two speakers born outside South America now live there; both are former Soviet citizens working in Brazil. Of course, the numbers are so low that one should take care not to see patterns that are not really there. (The same goes for Africa – there are two speakers from there, one working in Africa and one in the States.)

The article states that there are four different ways to view geographical shifts. In its words, these are (a) individual freedom, (b) the progress of science, (c) competition (as a way to advance science), (d) brain drain.

These are not really fully parallel to each other. For instance, “individual freedom” is not really a way to evaluate what goes on, or even a way to decide policy goals; rather, it would seem to be a principle that limits what policy tools we are willing to consider. At the same time, Andler states under this heading that “there are other compelling reasons to want to leave one’s country, e.g., miserable economic conditions or completely inadequate working conditions”. This belongs under its own (central) heading – namely, the conditions that make an individual able to work as a mathematician.

The second perspective (“the progress of science”) is said to be the one that what matters is the advancement of mathematics – and that it is our collective duty to ensure that mathematicians can develop themselves and work, and our individual duty to devote our lives to advance mathematics. This I would say to be uncontroversial, as far at least as mathematicians are concerned; not only is it something many could subscribe as a guiding philosophy – it is also an entirely reasonable way to frame the entire discussion. Other parties may have other goals in mind (national prestige, say, or, in the case of funding agencies, some more or less arbitrarily set formal goal), but most of the motives we would actually consider, including completely altruistic ones, fit nicely within this framework.

(Note that the article quotes Weil on dharma here. This is an example of something unfortunate: the clearest statement of a position is made by one of its more extreme proponents, and that, of course, has the effect of making the position seem a little less tenable.)

The third perspective – namely, “competition” – states that only by competing for the best faculty and students will universities have an incentive to keep or increase their level and give to faculty and students the working conditions they need to do mathematics. All of this is true, though one thing is not addressed – namely, that it is doubtful that, at the top of the pay scale in a few countries (US, say), further increases in salaries are really improving the ability of mathematicians to do mathematics, as opposed to simply serving as a tool for universities to compete (and a factor by which a few universities have a large, in-built advantage). It is also the case that salaries and especially working conditions can be set more by tradition than by anything else: for example, in the French system, salaries are essentially uniform regardless of location (thereby making faculty at some top institutions less well paid, in real terms, than in the provinces) – whereas, in the US system, which arguably has the largest financial basis of any, positions with truly light teaching loads are very rare, and positions with no teaching loads are essentially non-existent (in comparison, France’s CNRS opens up every year several positions with no teaching for life). Of course, part of the issue here is how to convince the body hosting the researcher that having the best students, or the best researchers, is really the priority; this is evident for us, but the financial source may have other goals in mind.

Lastly, we come to the “brain drain” heading. This is stated in the following terms: countries from where people emigrate lose the investment they made in their education, and they also lose the potential for further development; wealthier countries benefit – and also neglect making necessary investments in their own primary and secondary education; “it is much cheaper to import partially or fully trained young people”.

We have to look at this issue in the light of the data above.

(a) There is one very clear case of massive migration of people with PhDs from one place, namely, the former Soviet Union; this has to do with the implosion of an entire country. (We also see that many speakers left the rest of Eastern Europe already before the PhD stage.) Other than that, what we see is that large numbers of future speakers from outside the US did their PhDs there, but that the net flow to the US after the PhD stage was actually negative.

(b) As far as Latin America (say) is concerned, the issue is not a large net outflow (there turns out to be barely any) as low overall numbers. The same is true of other developing areas. It is striking that there are no speakers from India, given its mathematical tradition. (As for East Asia, it is difficult to reach meaningful conclusions, given that the Congress is in that geographical area this time around.)

Let us make our focus a little more precise. The article mentions some arguments for and against migration; as it states, they sometimes do not apply well to mathematics, whether they are under the ‘for’ heading (it is hard to see how (to use a paraphrase) “migrants sending money back home” is relevant here – though there is an analogue, namely, those cases where somebody from X manages to obtain substantial political power in the academic community in country Y, and uses it to procure funds to develop mathematics in X) or the ‘against’ heading (is having top mathematicians work full-time in a country really something that will improve significantly the teaching of students who do not intend to become research mathematicians? – the article seems to assert this).

As for costs saved by the USA (say) on education – figures per capita can be misleading here. Figures such as “$142,000 total average expenditure per student in primary/secondary education” are obtained much like similar figures on how much a prisoner costs: the total cost of a system – much of it consisting of fixed costs – is divided by the number of students or prisoners, as the case may be. What would be relevant here is not so much the marginal cost of educating an additional student, but the cost of having a better primary and secondary education system (or the cost of programs to supplement basic education). As far as the investment that the country from which the emigrate can be said to lose – obviously, what the state invests per student is often much less in developing countries, and not all students are supported by a public education system; rather, we could speak of what a country loses (to proceed with the same sort of logic) by not investing on a working postgraduate education system.

Still, these figures can be conducive to the right picture – namely, the academic system in the United States rests to a large extent on people who got their basic and undergraduate education elsewhere. The tables in the article should be enough make that clear. An awareness of this reality could, and should, contribute to create a common sense of responsibility. (On the French side, say, it should also contribute to create a sense of possibilities.)

Let us then restate the main issue within a clearly defined framework. There are young people with a great deal of talent and interest in mathematics in every part of the world. How do we ensure that they can develop their talent fully, and put it in practice to the best of their ability?

“We” here means anybody in the world who has an interest in the development of mathematics, or who considers wasted talent a pity and a waste. The way that we are phrasing the question sets certain perspectives deliberately outside its focus – namely, those based on national prestige, or “return on investment”. At the same time, lest the focus be thought of as narrow, let us emphasize that the question should not be thought of as concerning only an individual in the short run.

Consider, within this perspective, a system whereby talent is nurtured effectively in all countries, developed further in a few, and then put to work wherever it might be the case. Such a system would be a fair solution if the chances given to all students, regardless of origin, were equal or nearly equal; it would be a feasible solution if, and only if, it were sustainable. It may not be the best solution, let alone the only conceivable solution. However, it would be, for us, within the set of admissible solutions, provided that these two crucial “if”s are satisfied.

A few brief notes to supplement the above. We are talking about research mathematicians here, and not, say, about physicians and secondary school teachers, whose retention is a different issue altogether. (This is not, of course, to say that the issues raised by Andler on “adequate working conditions” and “miserable economic conditions” would not apply there.) We may even specify “leading research mathematicians”; this is, after all, what the database on ICM speakers is about.

Second – while the focus above may not be exactly the same as that of, say, government agencies that fund or could fund mathematics, this does not mean that the goals are all that different. Even from the viewpoint of “national prestige”, almost all would now agree that it is better for a country to produce very good football players (say) that work abroad, rather than not to produce them at all. It is also the case (for mathematicians or football players) that a country’s education system may be credited to the extent that it actually contributed to a professional’s formation; people do notice this – and thus it may make little sense, from the viewpoint of prestige, to see an exit from a country’s system at the bachelor’s or doctoral level as a greater loss than an exit that happens earlier.

Lastly, since the discussion may centre on mathematics in developing countries, let us give some examples from middle-income and high-income countries to clarify the framework of the discussion. An exodus of the proportions of the one that happened around the collapse of the Soviet Union is clearly something that gives rise to a non-sustainable situation. (Many would call it an effect of a non-sustainable situation as well, particularly given academic salaries in Russia in the early 90s.) The level of the country’s system for producing young mathematicians must clearly suffer as a result of such a shock ( – and as a result of the same drastic shortfalls that gave rise to it, some would add).

A somewhat different example is that given by the case of Germany. Here, again, the tables confirm what we already thought we knew: in net terms, Germany loses people after they get their doctorates. This is so in spite of senior academic salaries that compare favorably with those in large parts of Europe. The conventional guess – which is probably correct – is that this is due to a structural problem: Germany has nothing like tenure-track or associate professorships, or postes de maître de conférences; there are temporary “collaborators” and then there are full professorships. This is a problem that will not concern us here, at least as in so far as PhDs from Germany seem to be able to find jobs elsewhere. At the same time, it is a kind of problem that would legitimately concern some people in Germany, in that the system would be able to retain more people, and attract some, if it were structured differently. The same goes for any other country in a similar situation.

At this point – with a definition of the problem and its scope – we are at the beginning of a meaningful discussion. I thought briefly about the possibility of sketching the situation in a South American country (say). I may still do so soon. However, if you have read so far (congratulations!), you probably agree that this is a good point at which to declare the discussion open, and to hear what people have to say about (a) the situation in their own home countries, or in countries they are acquainted with; (b) how we could become better at recognizing and developing mathematical talent, at a global level; (c) the same, on placing mathematicians; or rather, how, given current trends in geographical shifts after or before the PhD level, we can still find viable ways to go much further on (b), even when this is far from completely apparent, and even when this goes against an overly simplistic take on “brain drain”.

Here is the website.

We are still putting together funding, but we have already managed to ensure quite a bit, and so we will be able to fund quite a few graduate students and young researchers (and perhaps some that are not quite so young). Our aim is to cover the expenses of admitted applicants in South America fully, and to cover the local expenses of people from elsewhere as well.

I’m not calling it a “summer” or “winter” school, since climate at a moderate-high altitude (3400 meters over sea level) relatively close to the equatorial line simply does not follow that categorization. “Dry-season school” would have sounded a little odd, even though it would have been accurate.

(Implications: sunny, cool at night and in the early morning, no mud, no rain, and unfortunately, no mushrooms either.)

As I said: people of all genders are encouraged to apply – or, in Spanish, tod@s están invitad@s a postular, which is a rather nice way to put it, since it explicitly includes cyborgs.

I hope the speakers will find they have been fairly depicted by their photograph:

This depicts the use of the Monte Carlo method to approximate an area. Many thanks to Martín Chambi!

At the same time, even though I had many reasons for staying away for blogging, it is a bit of a pity that I did not have time to blog precisely at a time when there started to be many more potential readers for it. Of course, some of them (you) are probably still around.

So far, I have written here mostly on mathematics, cinema and my travels. I would also like to touch a bit more frequently on a few serious subjects, not strictly mathematical, though sometimes related to mathematics and academia.

There is an idea going around my head since I read T. Rothman’s Genius and Biographers: The Fictionalization of Evariste Galois. This well-received essay from 1982 has had a definite influence on how Galois is seen nowadays; see a summary by a popular science writer, or the references in the introduction to the third edition of I. Stewart’s Galois-theory textbook. It has, to a great extent, replaced the account in E. T. Bell’s rather dated Men of Mathematics (1937). (E. T. Bell’s book is a collection of short biographies that inspired generations of mathematicians, while being famously imprecise and slanted, to say the least.)

It’s easy to correct Men of Mathematics on just about anything; at some point, before its general lack of accuracy was known, it may also have been necessary and worthwhile. What is, then, genuinely bothersome – or simply wrong-headed – about Rothman’s article?

First, it comes across as an effort, not just to defictionalize Galois, but to deromanticize him. The two concepts are not identical. Romanticization, strictly speaking, consists in the projection of a sensibility or the incorporation into a narrative, rather than in the practice of playing fast and loose with the facts. More importantly, there is no projection here, in the sense of imposing a sensibility alien to the subject. Galois lived in a romantic age; to understand his behavior, we must accept that hero worship, the search of sacrifice and martyrdom, the simultaneous identification with the people on the part of progressive sectors of bourgeois youth – combined with claims on a previously aristocratic concept of honour – were all concepts that pervaded the climate, rather than parts of a later grid. Rothman states under the heading “Harsher words” that “[Infeld] intends to make Galois a hero of the people”; this is a rather odd indictment, given that Galois was a member of an illegal organization called La societé des amis du peuple. We can discuss to what extent Infeld’s (socialist) conception of le peuple differed from that of Galois’s friends, or mention, as others have done, that (both in Galois’s and in Infeld’s time!) many of those self-identifying as amis du peuple were of middle-class extraction; still, what cannot be nullified is Galois’s self-inscription into a developing collective narrative.

Is it healthy for a biographical article to be written in a way that is so out of tune with its subject’s sensibility? This is not necessarily a disqualification; what is odd is to view accounts closer to an emic perspective as imposing an alien narrative. An etic perspective may differ sharply from an emic perspective; an etic account can draw attention to this fact without invalidating itself — and, conversely, it should not confuse the presence of a particularly large difference with objectivity.

To give an extreme analogue: an atheist may of course write a biography of Joan of Arc — and a historical essay on her should certainly be written differently than whatever account of her purported miracles was used at the Vatican for her canonization. Still, if a biographer charged another with “intending to make Joan of Arc into a religious figure”, something would be seriously amiss. We would also nowadays be careful not to rush to pathologize whatever seems to us unusual in some of her narrated experiences, given that, say, to state that one had visions was seen as acceptable and in consonance with the sensibility of a sector of society at the time. We would try to contextualize matters, even though we can probably agree that it is easier to make a case for a pathology in her case than in that of Galois, who had no visions and scaled no walls. Calling either self-destructive (not Rothman’s term) is to both say a triviality and to miss the point; it was neither’s intention to maximize his or her chances at survival.

Evariste Galois at 18, as drawn by a classmate.

But Galois’s troubles were not yet over. A few days later, he failed his examination to l’Ecole Polytechnique for the second and final time. Legend has it that Galois, who worked almost entirely in his head and who was poor at presenting his ideas verbally, became so enraged at the stupidity of his examiner that he hurled an eraser at him. Bell records this as a fact but according to the little-known study of Joseph Bertrand the tradition is false. Bertrand, who appears to have detailed information about the event, records that Galois, while expounding on the properties of logarithmic series, refused to prove his statements to the examiner M. Dinet and, in response to Dinet’s questions, replied merely that the answer was completely obvious. So was the result. [my highlight]

Rothman does not mention a version mentioned by Stewart (Galois theory, “Historical introduction”, 1973):

A variant asserts that Dinet asked Galois to outline the theory of “arithmetical logarithms.” Galois informed him, no doubt with characteristic bluntness, that there were no arithmetical logarithms. Dinet failed him.

(Stewart then goes into an interesting digression, stating that it is possible that Dinet might have been referring to the index modulo . This seems unlikely at first sight: the term “discrete logarithm” for what Gauss himself calls the index sounds like much later nomenclature — and, since Disquisitiones Arithmeticae was still relatively recent and Dinet has passed to posterity mostly for failing Galois, it does not seem plausible that Dinet would have been thinking of this. I will gladly stand corrected on this, however; where can one find what was expected from a candidate to admission at École Polytechnique at the time?)

At any rate, the meaning is clear. Rothman (quite reasonably) rejects Bell’s claims of eraser-hurling (which, according to Stewart, go back to Dupuy); then, he gives a version in which there is no misbehavior on Galois’s part, but simply some impatience at an imprecise remark. It is not extraordinary that a second-rate individual would have perceived Galois’s attitude as petulant. What was perceived as extraordinary even by Galois’s contemporaries is that such an individual would have then taken this as a sufficient reason to fail the candidate.

Rothman very nearly comes across as taking the opposite view. “So was the result” does not just make it seem as if the outcome should have been expected (by a seventeen-year-old candidate); it is a kind of statement that, by taking a response on the part of a individual up in the hierarchy as if he were a force of nature, manages to condemn the person down in the hierarchy, while pretending not to pass judgement. This kind of shorthand should be familiar enough to all readers; we are Rothman’s contemporaries. (Rothman quotes Galois himself as stating “Hierarchy is a means for the inferior”; he seems to have little time for such sentiments.) Rothman later says that “[he] do[es] not wish to suggest Galois should have been failed.”; if he had not already come close to suggesting as much, such a disclaimer would not have been necessary.

There is an entire theme to be developed here: neither Rothman nor romantic “historians” are indulging in anachronism – rather, Rothman would seem to sympathize with the hierarchical sensibility that condemned Galois, and that still exists in some weakened and modified form to this day. (An actual continuity here may be a point for debate; it may be simply a case of one hierarchy’s sympathy for another, with which it identifies.) What would have been impossible in the early 19th century is something else, namely, Rothman’s amateur psychologizing. This and the defense of academic hierarchy are related, however, in that the sort of superficial and conventional “psychology” in which Rothman’s essay engages is precisely the sort that is used nowadays (or, in many places, a generation or two ago) to defend a hierarchy, while implying that only an individual can be unreasonable.

Rothman’s main imputation is — no surprises here — is that Galois had “developed not a little paranoia”. At some point, academic paranoia will be understood by all to mean something rather different from the common kind – that is, it will be a set phrase imposed by force of habit. In the meantime, however, paranoia is still a clinical diagnosis, and, unless it is meant as an insult (much like, say, the originally medical term “idiot”), it has to be supported when used to describe a scholar much as in any non-academic context.

Some of the evidence adduced is decidedly odd. A shot was fired from a guard’s garret into a cell that Galois shared with several other prisoners. This was interpreted by Infeld as an attempt on Galois’s life. Rothman says he has “tried to present this episode in as neutral a tone as possible”; apparently, the way to do this is to seem to go to some length to attempt to defend the decision to throw Galois into a dungeon (“evidently because he had insulted the superintendent”) together with the man who was actually shot – something that was considered by other political prisoners to be unusual and completely out of line. At any rate, we are given no evidence that Galois believed that the shot had been aimed at him; the mere belief that a shot fired into a prison cell may have been intentional is enough.

The other evidence is that Galois took rejection letters badly. There is also another little matter – at least one of his manuscripts got lost after submission to the Academy. (Whether a second memoire got lost or neglected by Cauchy is something that seems unproved in either direction; Rothman cites R. Taton’s case against this – and also gives references that suggest that the suspicion of intention on Cauchy’s part was solely Infeld’s, and not Galois’s.)

Rothman’s essay ends in a sardonic note:

The underlying assumption is apparent: Galois was persecuted because he was a genius and all scientists, to a greater or lesser degree, understand that genius is not tolerated by mediocrity. A genius must be recognized as such even when standing drunk at a banquet table with a dagger in his hand. […] This is a presumption of the highest arrogance.

In fact, some of the material there and elsewhere gives a picture of a young man who was generally known to be, at the least, very talented; word of this had got around in academic circles, and also beyond that – he was “our little scholar” to other prisoners. The likely reasons for his sad and brief life and career can be multiple – but we cannot say that he had somehow managed to make his talent unrecognizable.

Évariste Galois, as drawn sixteen years after his death (1848) by his brother Alfred.

Rothman’s essay was brought back to my mind by an essay by M. Duchin (marked as juvenilia on her webpage). There, Rothman is paraphrased as having shown that E. T. Bell changed the chronology of events; it is also stated that Galois’s father’s suicide helps to explains the result of his (oral) examination at the École Polytechnique immediately thereafter. In all fairness, Rothman states that Bell’s main source does not make clear the chronology of events; moreover, Rothman can be interpreted to mean that Galois was in a particularly irritable mood (something that, in his view, makes claims of “the examiner’s stupidity” less valid), rather than to insinuate that Galois did badly in some objective sense.

In general, I found Duchin’s essay thought-provoking, and I certainly share her strong suspicion of the concept of “genius” itself. Still, I was unconvinced by her contention that there is something particularly male about genius-worship. I had also thought that there had been a transition at some point between the early nineteenth century, when one could speak of the genius of somebody (originally something close to a not entirely beneficent daemon), to popular usage in the twentieth century, when it became extremely common to say that somebody was himself (or sometimes herself) a genius. This arguably crucial shift is left unexplored.

Since I am travelling, here are the obligatory tourist photographs.

The sea

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