What is the mathematical genealogy of Felix Klein?

[Chris Hillman]

For another BRS thread, I looked up some information at http://www.genealogy.ams.org/ and was completely sidetracked by my attempt to look up the genealogies of my undergraduate mentors

Seriously, physics needs a tool like this, because it really does give amazing insight into the natural line of development seen in mathematics when you know enough to pick out the threads.

The graph of "geneological descent" rapidly becomes utterly unwieldy, but I'll try to keep a few partial graphs in this thread, with some explanatory glosses. The full graph is not a tree in the sense of graph theory, because some people have two supervisors, which rapidly tangles up doctorvater relationships into a complicated network.

It is interesting to see that certain persons function as highly connected nodes in the graph. Some of the best known names turn out to have thousands of descendants--- Carl Friedrich Gauss, for example, has almost 50,000 mathematical descendents. Other lines die out immediately for sad reasons, e.g. Niels Abel (died too young to have any students). A handful of persons, such as George Green, who are now regarded as great mathematicians don't appear at all because they were amateurs who never earned a formal Ph.D. And if you go back far enough, the academic system as we know it (in which a professor has doctoral students) simply had not developed. And some differences between different systems, e.g. "imperial" Germany, France, the UK, the Soviet Union, and the U.S. become apparent.

It's also interesting to see that leading mathematicians are often biologically related--- it's no accident that as you trawl through these trees you will see certain family names multiple times. Mathematical marraiges can reveal natural lines of development in the history of mathematics. Just to add to the fun, some of the people who appear are also biologically related to certain persons whom I would have to characterize as... dissidents.

If you know about youthful friendships and who was a post-doc of whom, you can learn even more. E.g. Emmy Noether was a post-doc of David Hilbert in Goettingen, where other great mathematicians on the faculty included Felix Klein and Hermann Minkowski. And Klein and Lie were student friends who greatly influenced each other. Eventually I hope to tie this all into explaining how "Noether's theorem" as known to physicists is a pale shadow of the actual result, which belongs to Klein's theory of symmetries of differential equations (the ones generated by a Lagrangian).

Seriously, this is a rich resource for anyone interested in the history of physics (because so many mathematicians played a crucial role in physics via their mathematical insights).

 

[Chris Hillman]

BRS: Mathematical Geneologies: Sophus Lie

The descendants of Sophus Lie include

• the greatest geometer of all time, perhaps, Elie Cartan
• Charles Ehresmann, best known for the notion of Ehresmann connection and the theory of fiber bundles, which is fundamental in modern fundamental theoretical physics,
• Fields medalist Vaughan Jones, best known for his work on knot polynomials (with connections to von Neumann algebras, and also to the work of Louis Kauffman which relates this stuff to statistical mechanics).

He and his friend Felix Klein visited Paris in 1871 (there is much more to this story, which I have told elsewhere), where they were greatly influenced by the great French mathematician Gaston Darboux, whose many students included

• Emile Borel, whose students included
  • Henri Lesbegue (best known for the Lesbesgue integral),
  • Paul Montel (best known for Montel's theorem in complex analysis)
• Emile Picard (known for the Picard group in the theory of functions and his fundamental theorem in the theory of ODEs),
• Thomas Stieltjies, best known for the Stieltjies integral,
• Eduard Goursat, best known for the Cauchy-Goursat theorem in the theory of functions of one complex variable
• Elie Cartan again, because Cartan had two supervisors.

Darboux descends in turn from Simeon Dennis Poisson (best known today for the Poisson equation and the Poisson integral formula in the theory of harmonic functions) through Michel Chasles, and via Poisson is connected to Lagrange, Dirichlet, and Liouville, all of whom are connected to mathematicians too numerous to name.

Sophus Lie was a student of Carl Bjerknes, whose biological descendants include Vilhelm Bjerknes, whose Jungentraum became a reality with the rise of computer modeling of weather systems--- and a certain anti-Einstein fanatic Carl Bjerknes was a student of Bernt Holmboe, who was the supervisor of Niels Abel.

Few physicists, I think, properly appreciate the breadth and depth of the mathematical work of Darboux and Minkowski, in particular. Some day I hope to try to explain.

Here is a graph (incomplete in both directions, and including only a selection of students of the mathematicians belonging to some vertices).

 

[Chris Hillman]

BRS: Mathematical Geneologies: Felix Klein

Felix Klein, one of the historical figures who has most influenced my own view of mathematics, has a huge number of descendants (see graph below); here is a brief gloss:

• Ferdinand Lindemann is best known today for his proof that pi is transcendental;
• David Hilbert was, with Henri Poincare, the last polymath; he is remembered today for achievements too numerous to list, but I'll mention his work in invariant theory, foundations of mathematics, his highly influential championing of general relativity, and his famous Hilbert Problems, which give some indication of the depth and breadth of his interests. His many mathematical descendants include:
  • Hugo Steinhaus was a key member of the Polish school (one of the tragic discoveries in these graphs is the cost of World War II; Steinhaus was one of the victims of the Original Nazis)
  • Richard Courant, a leading analyst eventually fled the Nazis to Columbia University, where he had many descendants; some of you probably know his multivolume textbook on mathematical methods,
  • Richard Brauer is one of the central figures in the theory of linear representations of groups,
  • Heinz Hopf was one of the most influential topologists (c.f. Hopf fibration)
  • Eduard Stiefel is best known today for Stiefel manifolds, which I hope to discuss one day,
  • Friedrich Hirzebruch is known for extremely influential work in numerous fields which led to among other achievements the Atiyah-Singer index theorem, one of the most famous results of 20th century mathematics,
  • Saunders Mac Lane founded category theory with Garrett Birkhoff (son of George Birkhoff, known to mathematicians for Birkhoff ergodic theorem, and to physicists for a fairly trivial result on spherically symmetric solutions of the Einstein field equation and perhaps for his long defunct theory of gravitation, which some dissidents in the physics fringe mistakenly believe is still viable today),
  • John Thompson is best known for his work on group theory, which was essential for another monumental achievement (perhaps) of 20th century mathematics, the classification of finite simple groups,
  • Irving Kaplansky was a leading algebraist in the Chicago School c. 1950-1960 (sometimes called the greatest math deparment of all time)
  • David Eisenbud is a leading algebraic geometer whose textbook Commutative Algebra with a View Towards Algebraic Geometry I often cite,
  • Hyman Bass is a leading mathematican known for his work in K-theory (c.f. the index theorem)
  • Alex Rosenberg is best known for his invention (with Stephen Chase) of a Galois theory for rings.
• Hermann Minkowski is remembered by mathematicians for the Geometry of Numbers and by physicists for inventing the notion of spacetime; if you know
  • HM and his colleagues were interested in Lie's theory of the symmetry of PDEs,
  • the Moebius group in the theory of linear fractional transformations is SO(1,3),
  • the "point symmetry group" of the Laplace equation is the conformal group of R^3, namely SO(1,4),
  • Lie and others had proposed Kleinian geometries which were very close to Minkowski spacetime,
  • Minkowski numbers, which arise in studying the wave equation, lead directly to metrics with (1,n) signatures
  this discovery seems much more natural! (among the less well known achievements of Klein and Minkowski is work on multidimensional continued fractions which didn't make it into my own diss on generalized Penrose tiling spaces); his descendants (few because he died young) include:
  • Constantin Caratheodory is best remembered by mathematicians for his work in real analysis, and by physicists for his formulation of classical thermodynamics,
  • Paul Finsler's theory of non-Riemannian geometries has recently become popular among physicists, almost a century after he invented it.

Figure: some mathematical descendants of Felix Klein (highly incomplete graph!)