Marcel Berger

See item 5 below for a prize.

Math Reviews of systolic papers by M. Berger

A la recherche des systoles, by Marcel Berger

This page contains Marcel Berger's personal story concerning systoles, as told by the master in the summer of '05. We have limited editorial changes to a bare minimum, so as to preserve the flavor of the original account.

1. Thom on Blatter. ``To the best and fairness of my knowledge, here is my recollection of my relations with systoles. ``For me things started in the very small [room of the] Strasbourg Mathematics Department Library, I think in the fall of '61 or in the spring of '62, I cannot remember [exactly]. [René] Thom was there, looking at the newly arrived journals, I was close to him in the small room. Of course he was my hero. Then he came upon the Blatter paper (Blatter '61), and spoke to me, saying, `You know Berger (still now most French mathematicians call between them or quoting their [last] names not by the first name), this paper is very interesting because it concerns the general problem to find universal (independent of the metric) relations between the volumes of various homology, etc. classes.'

``Coming from Thom, I considered that question as good (and natural also, which I like). I started looking at various generalizations. I was really getting nowhere. The only thing I did was to learn [P.M.] Pu's proof, and then to make propaganda for the topic in my Bombay lectures (Berger '65).

2. Chern, Wirtinger inequality, and calibration. ``I then remember talking about Thom's problem with [S.S.] Chern (I think in Berkeley where I was spending the summer quarter '68 or the summer of '69, I am not sure) about the question. He told me two things, the first was that his feeling was that such a universal inequality could be true in the middle dimension (for even dimensions). The second thing he did when thereafter I asked him about the complex projective space, since I was trying to look at Pu generalization to other projective spaces. He told me for the complex case that this followed from Wirtinger inequality (I did not know Wirtinger stuff at that time).

``I kept working with the topic, and I could really get nowhere. The only thing not trivial I could prove was that the inequality was OK for the quaternionic projective space and the Cayley plane (for their standard metric). To do this I had only to prove that the 4-form (respectively, the 8-form) was what is called today calibrating after (Harvey and Lawson '82). The quaternionic case was easy [see item 5 below], but I had a hard time to prove the Cayley case. To make a living, I published the two papers (Berger '72a) and (Berger '72), but you perfectly know that there is nothing in them, except propaganda for the subject, and the two calibrations. Then I [made] no contribution, except having kept making propaganda. You can look at the wonderful introduction book (Morgan '88-'00), page 75, section 6.5: history of calibration. I kept making propaganda, in lectures, and by some writing (in part).

``Then came [M.] Gromov, and his followers; I still continue propaganda, but as you know the subject started soon to be popular.

3. Three notes. ``The first is the explanation for the two `funny' titles, in case you are not familiar with the famous French novel `A la recherche du temps perdu', by Marcel Proust. The year '71 was Proust centenary, so I used that pretext to imitate the first two sections of his novel which are called `Du côté de chez Swann', and `A l'ombre des jeunes filles en fleur'.
``The second thing is the origin of the word "systole", in case I have not already told you. I was looking at [the] time for a word of the type `iso -???-ic' both for the systoles and for the injectivity radius. I looked at Greek language dictionaries (French to Greek, basically there are no good ones) and found various wording. [Luckily], I was doing physical education together with a Greek literature colleague; he told me what I was proposing was `low Greek', of course I explained to him, in ordinary words, what were a systole and the injectivity radius. The next week physical education session he came back with the two proposal: "isoclysteric" and "isosphincteric". I was still young in the bad sense, say provocative, I was [amused]; but I told these two wording to Besse seminar next week, they were horrified and told me "Marcel, you cannot use that". So at the next session I asked him to find a less bad taste "ic" stuffs. Next week he came back with "isosystolic" and "isoembolic". Seminar people were happy; you understand that in short I switched from proctology to cardiology. Both wording were immediately accepted by Gromov.
``The third thing is more important. In the same library, Thom asked me another question: `Berger, I give you a compact manifold: does there exist a preferred metric on it?' This started my propaganda for Einstein manifolds. So that in conclusion, after my dissertation and the pinching stuff, all my interests were given to me by Thom.''

4. Bibliography. The above account refers to the following texts.

Berger, M. ('65). Lecture on Geodesics in Riemannian Geometry, Tata Institute of Fundamental Research.
Berger, M. ('72). A l'ombre de Loewner. Ann. Scient. Ec. Norm. Sup. 5: 241-260.
Berger, M. ('72a). Du côté de chez Pu. Ann. Scient. Ec. Norm. Sup. 5: 1-44.
Blatter, C. ('61). Über Extremallänge auf geschlossenen Flächen. Comment. Math. Helvetici 35: 153-168.
Harvey, R. and Lawson, H.B. ('82). Calibrated geometries. Acta Mathematica 148: 48-157.
Morgan, F. ('88-'00). Geometric measure theory: a beginner's guide, third edition in '00, Academic Press.

5. Berger prize. In the hallowed tradition of P. Erdös, we are hereby offering a prize of $50 for a calculation of the middle dimensional optimal stable systolic ratio of the quaternionic projective plane, see explanation. For European contestants only, the prize is upped to €50. Weigh also other prizes.

Further historical remarks may be found in the entries under Loewner and Rodin. Consult a history of systolic topology.

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