One of the more nerdy web resources available to mathematicians, after the collaboration distance calculator on MathSciNet, is the Mathematical Genealogy Project. This contains a lot of data on the PhD supervisors or students of mathematicians. A lot of fun for browsing; one of my PhD students, a geometer, was delighted to find the name of Oswald Veblen among his ancestors.

The MGP used to trace my ancestry back to Erhard Weigel, the supervisor of Leibniz. (In fact Weigel, one of the most prolific people in the MGP database in terms of mathematical descendants, was a lawyer!) However, new data has been added, allowing multiple advisers, so that rather than a single line of descent there is now a branching tree, including such luminaries as Luca Pacioli and Nicolaus Copernicus. Some lines go back to the late fourteenth century now. I haven’t explored everything here; hours of fun ahead!

Of course, a resource like this is no better than its data. Among the less serious difficulties, here are a couple in my own ancestry. Even the MGP recognises a problem: Lagrange is listed as a student of Euler: as they put it, Lagrange had “No dissertation, no advisor, but we show a link to Euler to show a connection in our intellectual heritage.” The History of Mathematics site says of Lagrange that “largely he was self taught and did not have the benefit of studying with leading mathematicians”, though he corresponded extensively with Euler.

Then we come to Henry Whitehead, listed as a student of Veblen. Peter Neumann once told me that Whitehead went to Princeton to work with Veblen, but was not formally his PhD student. (However, the History of Mathematics site agrees with the MGP that Whitehead did take his PhD at Princeton under Veblen’s supervision.)

I am reminded of what Béla Bollobás says about Littlewood in the introduction to *Littlewood’s Miscellany*:

… a mathematical proof containing gaps reminded him of being descended from William the Conqueror — with two gaps.

There are some larger problems with the MGP data. Some of these:

- The PhD as a formal qualification is, essentially, a 20th century invention; so whatever the earlier links mean, they is not as precise as more recent data.
- It is very common today for a student to have two supervisors; the MGP allows this, but it is not clear whether it is (or should be) enforced.
- Perhaps most seriously, the MGP relies on information sent in by individuals, so can make no claim for completeness or even accuracy.

None of this matters for a student curious about her pedigree; but if anyone were to try to use this data for scientific purposes, it might matter.

I mention this because someone has done just that. In last week’s *Nature*, there is a paper entitled “The role of mentorship in protégé performance”, by R. D. Malmgren, J. M. Ottino and L. A. Nunes Amaral.

After establishing that “mentor fecundity” (their term for the number of successful PhD students a supervisor has – I have to grit my teeth to write it) is strongly correlated with other measures of academic success such as number of papers or membership of the NAS (no control for nationality here, as far as I can see, even though the MGP website displays helpful little flags), they go on to other observations:

- mentors with low fecundity train protégés with higher than expected fecundity;
- mentors with high fecunduty train protégés with high fecundity early in their careers, but their protégés in the last third of their careers have relatively low fecundity.

None of this is explained, and none of the hypotheses tentatively suggested is supported very strongly by the data. This invites one to look for other explanations.

The first observation is unclear: how is “expected” calculated? Do we expect students to be just like their supervisors, or just like the population average? If the former, then “higher than expected fecundity” is exactly what you would expect; it is otherwise known as “regression to the mean”.

The second observation provides more of a puzzle. The authors assume that choice of a student or supervisor is more or less free. The fact that it isn’t could make a big difference. To take just one example, once someone is recognised as a “successful” supervisor, he may well be expected to take on potentially difficult students. On the other hand, a bright new member of department may be assigned a good student in order to help her career get off to a strong start.

Another possible effect is that the intellectual interests of a prolific mathematician are likely to be broader in late career than when starting out; successful students may move outside mathematics and be unreported in the database. In my case, the reverse was true: my second student, Eric Lander, has made a successful career in genetics; the MGP lists him as having only one student, Manolis Kamvysselis, whose thesis title is given as “Computational Comparative Genomics: Genes, Regulation, Evolution”. I know that Eric has had more than one student!

(When Oxford celebrated Henry Whitehead, a table of his mathematical descendants was prepared – this was long before the MGP began. Under Graham Higman was written “Here be algebraists”; clearly, for the compilers of the table, algebra was outside their comfort zone. Fortunately this doesn’t apply to the MGP.)

The advent of multiple supervisors is also likely to distort the picture (though the paper ends its sampling at 1960 and so is free of this). It might be expected that a student assigned to a young member of staff will be given a “safe pair of hands” as second supervisor.

There is also an issue which arises now with clinical trials: the invisibility of negative results. Students who do not complete a PhD (yes, there are some) never appear in the MGP tables, so their effect cannot be considered by such a survey.

As for myself, I have had many outstanding students. Moving from Oxford to Queen Mary in mid-career cannot but affect my “fecundity”, but without detailed knowledge you probably couldn’t tell whether it would have increased or decreased it. But I have learned far more from my students than I could ever hope to have given them. Surely that is what the relationship is really about.

But, no doubt, the bean-counters will take notice of research like this in their quest to make universities more “productive” at lower cost…