See item 5 below for a

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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