There is great public interest in genealogy. Many of us live in hope of identifying some illustrious forebear, or enjoy the frisson of having a notorious murderer somewhere in our family tree. Academic genealogies can also be traced: see this week’s That’s Maths column in The Irish Times (TM062, or search for “thatsmaths” at irishtimes.com).
A project called the Mathematics Genealogy Project (MGP) has assembled a huge database of mathematicians. Parent-child relationships are replaced by links between students and their doctoral supervisors or, before PhDs became common, by links between mentors and proteges. Usually this means one-parent families but in many cases there are two or more advisors, making the structure or topology of the family tree more complex and more interesting.
Of course, a true intellectual history implies much more than advisory links, with factors such as lectures, books, papers, correspondents and scholars who have strongly influenced a mathematician. But the mentor-protege connection catches much of the influence in a simple way. The database comprises a large graph, the mathematicians being the vertices or nodes, linked together by the advisory relationship.
To give an example, type in Grigori Perelman, the mathematician who in 2003 proved the Poincaré conjecture. Clicking back through the links, we get to Andrei Markov – Markov chains are central in the theory of stochastic processes – and, before him, to Pafnuty Chebyshev, known especially for his work in probability and statistics. Above him comes Nikolai Lobachevsky, famous for his development of hyperbolic geometry. Erasmus is also among Perelman’s forebears or “ancestors”. The trail eventually goes cold, but not until it has reached back to Manuel Bryennios at the beginning of the fourteenth century.
Since many mathematicians have more than one advisor, the number of forebears grows substantially as we trace further back, and the chances of having a famous ancestor are high. For example, Chebyshev has 10332 descendants. Still, it is great fun to find that Euler, Gauss or Galileo is in one’s academic family tree, even if about one in three mathematicians can make a similar claim.
Generally, the trail can be traced back for several centuries but eventually stops somewhere like medieval Oxford or Renaissance Bologna. Sometimes there are no forward links; for example, Descartes has no recorded descendants, so a direct link to him is impossible. Moreover, some prominent names are missing: there is no entry for William Thompson, Lord Kelvin. It may be argued that he was a physicist, but the scope of MGP is wide enough to cover applied mathematics and theoretical physics.
MGP was set up in 1996 by Harry Coonce, then at Minnesota State University. It was a labour of love, and he worked incessantly on developing the project, with only meagre resources. It is now hosted by the Mathematics Department of North Dakota State University ( MGP ) with mirror sites at the American Mathematical Society and in Germany and Brazil. The number of entries has grown steadily, with about 1000 new entries every month, and has now reached almost 200,000.
MGP has been criticized as an insubstantial project lacking academic gravitas, identifying links that have no particular importance or significance. Initially set up against some hostility and with little support, the MGP site is now recognised as a valuable resource. It is true that the mentor-protege links are not guaranteed to provide a meaningful chain between modern mathematicians and the ‘greats’ of earlier times. But as long as they are not over-interpreted, they do provide valuable information about the connections between mathematicians of different eras and locations.
If we assume that strong students identify good supervisors and excellent advisors take on only the best students, then by-and-large the better students should link back to the more famous mathematicians. A sparkling academic lineage may be only loosely coupled to academic prowess but, to be honest, we all get a thrill to link to someone famous, staking a claim to the glittering legacy of mathematics.
The MGP data base is like a giant genealogical chart, with the interconnections forming a network or graph. The vertices or nodes are the mathematicians, and the edges or links of the graph are the advisor/advisee connections between them. The graph is mostly like a tree, but it has multiple connections between vertices. Moreover, it cannot be represented by a diagram on a sheet of paper without some of the lines crossing. That is, the graph is non-planar.
We all know the problem of supplying electricity, water and gas to three houses without having any crossings amongst the connections: it is an impossible problem to solve. It involves the bipartite graph K(3,3) with two sets of three nodes, each of the first set being linked to each of the second set. The non-planarity of the MGP graph follows from a mathematical theorem of Kazamierz Kuratowski, which shows that any graph with a K(3,3) subgraph cannot be planar.
Prof Ezra Brown of Virginia Tech has found a subgraph K(3,3) in MGP, depicted below. The green vertices represent the “utilities” nodes and the yellow ones the “houses”.
The MGP database goes back to about 1300AD. The ultimate aim is to include all mathematicians in the world. The term ‘mathematician’ is interpreted quite broadly and includes, for example, computer scientists and theoretical physicists. Wouldn’t it be wonderful if we could establish a link back to Ancient Greece, and to the mathematicians of Babylonia, India and China. Who would not love to be a descendant of Archimedes? But – as Lewis Fry Richardson once said of computer weather forecasting – “that is a dream”.
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