Friedrich Hirzebruch
German-Russian Cooperation in Mathematics
von Friedrich Hirzebruch
Seit 1959 vergibt die Russische Akademie der Wissenschaften als höchste Auszeichnung ihre Lomonossow Gold Medaille, jeweils gleichzeitig an einen Wissenschaftler aus Rußland und einen aus dem Ausland. Zu den bishe- rigen Preisträgern zählen R. Mößbauer {1984) sowie S. Sobolev und J. Leray {1988). Im Jahr 1996 erhielten den Preis Nikolai Krasovskii und Friedrich Hirzebruch. Sein Festvortrag vom 29.5.97 während der Vollver- sammlung der Akademie in Moskau über Beispiele der deutsch-russischen Zusammenarbeit in der Mathematik seit der Gründung der Akademie im Jahre 1724 ist im folgenden in leicht gekürzter Form abgedruckt. (G.F.)
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11.03.97 • 59 FAX 8-10-49-228-402-277 Oear Professor Hirzebruch,
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Professor Criedrich Hirzebruch Max-Plank Institute of Mathemattes Bann, Germany
In the name of the Ru~Hdan Academy of Seiences we are glad to inform you 'that at its meeting on February 25, 1997 the Presidium of the Academy voted unanimously the following resolution:
«To award the M. Lomonosov Large Gold Medal of the Russian Academy of Seiences of 1996 to Professor Friedrich Hirzebruch
(Federal Republic of Germany) for his outstanding contributions to the field of algebraic geometry and algebraic topology».
According to its status, the H.Lomonosov Large Gold Hedal is the highest award of the Russian Academy of Sciences, and is awarded annually to two outstanding scientists - one from Russia and one from abroad.
This decision reflects our high esteem of your contribution to the mathematical science, your fruitful werk in the area of international science management, your contribution to the strengthening of cooperation between German and Russian. scientific communities. We congratulate you heartily with this well deserved award. 'iour numerous colleagues ·and friends in Russia join our congratulations.
According to the status of the M. Lomonosov Large Gold Medal, the ceremony takes place at an annual session of the Academy, where laureates deliver lectures (30-40 minutes) on a topic of their choice. This year the session of the Academy will be held on Hay 21-31. If this time frame is good for you, we will include the ceremony into the schedule, and will send you formal invitation immediate!~.
Once more our sincere congratulations,
+7-oG5-2372622 +7-oG5-237-Q107
(signature) 'iuri S.Osipov President (signature) Nikolai A. Plate Secretary-General Phone: +7-oG5-237-2622 Fu: +7-oG5-237-Q107
The academies in Berlin and St. Petersburg were founded under the influence of Leibniz. The German philosopher and mathematician Christian Wolff was an honorary member of your academy since its foun- dation and very active in building it up. There were attempts to malm him the first president. Lomonos- sow was born in 1711. He studied in Marburg in Germany under Christian Wolffand became a profes- sor of Chemistry of your academy in St. Petersburg in 17 45 and played an essential role in the founda- tion of the University in Moscow now named after him. I am proud to wear his image and to have the Lomonossow medal.
54
" · B. ,'I()"OIIOCOB 1711 1/(j.)
Leonhard Euler came to St. Petersburg already in 1727 where he died in 1783. Chrisitan Wolff wrote to him "You are travelling to the paradise of scientists
" Euler made your academy the world center of mathematics. His time in St. Petersburg was in- terrrupted by 25 years in the Berlin academy with Frederic the Great. Euler was a Swiss mathemati- cian, student of the Bernoullis in Basel. He estab- lished many ties between Berlin and St. Petersburg.
Nowadays your academy has the Euler International Mathematical Institute in St. Petersburg with many relations to Berlin and Germany, for example through the Berlin society of friends of the Euler Institute and
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through the Max Planck Institute for Mathematics in Bonn. I wish the Euler Institute much luck for the future.
The Gennan mathematician Christian Goldbach met Leibniz and Christian Wolf! in Germany in 1711, be- came the Secretary of your academy since its foun- dation, was an honorary member of the academy since 17 42 and worked for the Foreign Ministry ( cryp- tography, deciphering with great success, for ex- ample, an unfavourable message of the French am- bassador to his government about the reigning ern- press (Tsariza)). Approximately two hundred let- ters between Euler and Goldbach are available in the archives covering the period from 1729 to 1764 when Goldbach died, thus including most of Euler's Berlin time. Goldbach formulated mathematical con- jectures which gave work to famous Russian mathe- maticians up to now. What will happen to the inter- esting mathematical e-mail messages of present day mathematicians in 250 years from now?
I received the Lomonossow medal. Therefore I re- called with a few words the great time of the 18th cen- tury when giants like Lomonossow and Euler lived.
In 1990 I received the Lobatschewski Prize for my book "New topological methods in Algebraic Geom- etry". Your academy treats me too well. With Lo- batschewski we come to the 19th century. He de- veloped Non-Euclidean Geometry. Carl Friedrich Gauß at the age of 63 learned Russian to read Lobatschewski. Upon his recommendation Lo- batschewski was appointed corresponding member of the Göttingen academy. The Lobatschewski Prize was given for the first time to the Norwegian mathe- matician Sophus Lie, Professor in Leipzig, in 1897 for Volume III of his Theory of Transformation Groups.
Felix Klein in Göttingen wrote the offi.cial report on this book for the "Physiko-mathematische Gesell- schaft der Kaiserlichen Universität in Kasan".
With Felix Klein we come to our century. He pre- sented the work of Sophus Lie to the Congress of Mathematics held in connection with the world's fair in Chicago at Northwestern University, Evanston, in 1893. This congress was a prerunner of the Interna- tional Gongresses of Mathematicians which started 1897 in Zürich. The congress was here in Moscow in 1966. The next one is in Berlin in August 1998.
I take the opportunity to invite all those interested in mathematics to Berlin to strengthen the Russian- German and the International Cooperation.
My modest contribution to the cooperation consists in the following. Since 1956 I am Professor in Bonn.
I established the Mathematische Arbeitstagung in 1957, a meeting where the program is not determined in advance, but where the speakers are chosen by
German-Russian Cooperation in Mathematics
the audience in a general program discussion to en- sure that the most recent important developments are presented. Of course, I tried to have Soviet par- ticipants in this annual meeting. For well known rea- sons this was very diffi.cult. I always invited five times as many mathematicians from the Soviet Union as I had money in the budget. The financial risk was minor. But in June 1967 exactly 30 years ago we had great success. Five Russian colleagues attended.
The audience was eager to hear them. All of them were chosen as speakers, some even twice. Seven of the eighteen lectures of the Arbeitstagung 1967 were by our Russian visitors. This shows how much we profited, how much we were aware of the high inter- nationally leading level of mathematics in the former Soviet Union, how much we wanted to learn form them. The lecturers in the order of the talks were
I. Schafarewitch
D.V. Anosov
M. Postnikov Boris Venkov Yu. I. Manin
1) Simple Lie Algebras in Finite Characteristics
2) Algebraic Analogue of Uni- formisation
1) Dynamical Systems
2) Asymptotic theory of some partial differential equations K-theory for infinite complexes Cohomology of some groups On rational surfaces
The list illustrates high level and wide scope. By the way, the opening lecture was by M.F. Atiyah "Hy- perbolic equations and algebraic geometry". Atiyah spoke about the work of I.G. Petrowsky on the dif- fusion of waves and the lacunas for hyperbolic equa- tions.
Egbert Brieskorn, one of my doctoral students, was the first to discover (it was September 1965) that in complex dimensions greater than two the neighbour- hood boundary of an isolated singularity can be a
Friedrich Hirzebruch
sphere, exotic or not, opening an exciting new way to study exotic spheres. At the Moscow International Congress of 1966 I lectured about Singularities and Exotic Spheres. Let me mention that 8 years later at the International Congress of Mathematicians in Vancouver Brieskornset an example of German-Rus- sian Cooperation. V. Arnold was supposed to lecture on "Critical points of smooth functions". He was un- able to come. Brieskorn delivered Arnold's lecture from which all interested in singularities could learn so much.
Let me come to my Zürich teacher Heinz Hopf. He was a German who had received his doctor's degree under Erhard Schmidt in Berlin 1925. In 1931 he accepted a chair at the Eidgenössische Technische Hochschule in Zürich as successor of Hermann Weyl and escaped in this way the persecution by the Nazis.
He was of Jewish origin. From him I learnt some facts about the famous book Topologie I by Paul Alexan- droff and Heinz Hopf and how the close cooperation of Alexandroff and Hopf started. They met for the first time in the summer term 1926 in Göttingen.
Alexandroffreported about this in [A]: "Die Bekannt- schaft zwischen Hopf und mir wurde im selben Som- mer zu einer engen Freundschaft. Wir gehörten bei- de zum Mathematiker-Kreis um Courant und Emmy Noether, zu dieser unvergeßlichen menschlichen Ge- meinschaft mit ihren Musikabenden und ihren Boots- fahrten bei und mit Courant, mit ihren ,algebraisch- topologischen' Spaziergängen unter der Führung von Emmy Noether und nicht zuletzt mit ihren verschie- denen Badepartien und Badeunterhaltungen, die sich in der Universitäts-Badeanstalt an der Leine abspiel- ten. . . . Die Kliesche Schwimmanstalt war nicht nur ein Studentenbad, sie wurde auch von vielen Univer- sitätsdozenten besucht, darunter von Hilbert, Cou- rant, Emmy'Noether, Prandtl, Friedrichs, Deuring, Hans Lewy, Neugebauer und vielen anderen. Von auswärtigen Mathematikern seien etwa Jakob Niel- sen, Harald Bohr, van der Waerden, von Neumann, Andre Weil als Klies ständige Badegäste erwähnt."
Here one sees Göttingen (in particular a swimming pool) as a mathematical world center. (Alexandroff and Hopf liked to swim. In Dubna at the General Assembly of the International Mathematical Union in 1966 they asked me to join them swimming in the Volga. I had no swimming suit. I am still sorry I missed this unique possibility.)
As Alexandroff pointed out in his memorial article on Heinz Hopf, two Göttingen schools of this time ( the twenties) were especially prominent and active, one headed by Courant (Applied Mathematics) and one by Emmy Noether (Modern abstract algebra) both closely related to Hilbert.
Alexandroff was in Göttingen during all three sum-
56
mers 1926-28, Hopf during the summers of 1926 and 1928 and part of 1927. Emmy Noether attended their lectures. They learnt from her to base the ho- mology theory on group theoretical methods, not to study only numerical invariants of topological spaces (Betti numbers and torsion coefficients in the sense of Poincare), but to regard the theory, as we say today, as a functor from topology to algebra which asso- ciates a group or some other algebraic object to a space and a homomorphism to a continuous map of spaces. In the preface to their book [A-H] Alexan- droff and Hopf write: "Die allgemeine mathemati- sche Einsicht von Emmy Noether beschränkte sich nicht auf ihr spezielles Wirkungsgebiet, die Algebra, sondern übte einen lebhaften Einfluß auf jeden aus, der zu ihr in mathematische Beziehung kam. Für uns war dieser Einfluß von der größten Bedeutung, und er spiegelt sich auch in diesem Buch wieder. Die Ten- denz der starken Algebraisierung der Topologie auf gruppentheoretischer Grundlage, der wir in unserer Darstellung folgen, geht durchaus auf Emmy Noether zurück. Diese Tendenz scheint heute selbstverständ- lich; sie war es vor acht Jahren nicht; es bedurfte der Energie und des Temperamentes von Emmy N oether, um sie zum Allgemeingut der Topologen zu machen und sie in der Topologie, ihren Fragestellungen und ihren Methoden, diejenige Rolle spielen zu lassen, die sie heute spielt."
Heinz Hopf and Paul Alexandroff were winners of the Lobatschewski Prize in 1969 and 1972 respectively.
They deserved it for their book Topologie I.
Alexandroff and Emmy Noether were close friends. She came to Moscow for the winter 1928/29 to give a course on Algebra. In her letter of October 13, 1929, written exactly one year after her arrival in Moscow, she speaks about the growing interest in Topology and a meeting in Prague. She mentions van der Waerden's lecture at this meeting and his report on Pontryagin's dissertation.
Emmy Noether left Germany for the United States in 1933 after the beginning of the Nazi regime and the terror against the German citizens who were Jewish.
She went to Bryn Mawr College, lectured there and in nearby Princeton University. She died on April14, 1935. The address "In Memory of Emmy Noether"
given by Paul Alexandroff at a meeting of the Moscow Mathematical Society on September 5, 1935, ends with the sentence: "She loved people, science, life with all the warmth, all the joy, all the selflessness and all the tenderness of which a deeply feeling heart - and a woman's heart - was capable".
The meeting of the Moscow Mathematical Society honoring Emmy N oether was part of the first Inter- national Conference on Topology. The sensational new concepts and results would have been impossi-
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ble even to formulate without algebraic objects like groups and rings associated to spaces, via functors from topology to algebra. I will mention only two highlights: the cohomology theory including the ring structure (J.W. Alexander, I. Gordon, A.N. Kol- mogoroff) and the theory of characteristic classes ac- cording to E. Stiefel and H. Whitney (Hopf lectured on Stiefel's dissertation which concerns the charac- teristic classes of the tangent bundle of a manifold and applications. Whitney spoke about his theory of characteristic classes of a sphere bundle). Hopf reported on this conference on the occasion of the 75th anniversary of the German Mathematical So- ciety in 1965. The characteristic classes live in the cohomology ring of the manifold or more generally the base space of the bundle. How to distinguish an element in a cohomology group if one only knows the numerical invariants, rank and torsion numbers?
The characteristic classes have had numerous appli- cations later. For example, the role of Pontryagin characteristic classes in Thom's determination of the cobordism ringandin my work on the Riemann-Roch theorem. Frank Adams solved the vector field prob- lern for spheres using cohomology operations. They are natural operations between cohomology groups of different dimensions. How to define them if only rank and torsion numbers are treated? Lefschetz em- phasized at the conclusion of the congress "the great value that Emmy Noether's ideas had for the devel- opment of modern topology," quite a step from his point of view in his topology book of 1930 that the algebraisation is a mere question of a different termi- nology.
Emmy Noether was a pioneer in topology though her Collected Papers do not contain a single topologi- cal paper. However the concept of homology group appeared a little earlier and independent of her in papers by Leopold Vietoris in Austria who is living in Innsbruck. I just congratulated him for his 106th birthday to take place Wednesday next week.
Some years ago Albert N. Shiryaev discovered many letters in the datscha (the house in Komarovka) of Alexandroff and Kolmogoroff: Letters from Emmy Noether to Alexandroff (I mentioned one of them) and letters from Heinz Hopf to Alexandroff ranging from 1928 to 1968, interrupted by the war. They dis- cuss the work on their book (until 1935) and many other mathematical topics. In one of the letters of Hopf, I am very proud of it, I occur as a promis- ing young man. Then there are letters from Fe- lix Hausdorff to Alexandroff. By the kindness of Shiryaev I personally have copies of all these letters.
In the Bonn University archives we have letters from Alexandroff to Hausdorff. Indeed, the relation be- tween Alexandroff and Hausdorff is very dear to us
German-Russian Cooperation in Mathematics
Bonn mathematicians. Hausdorff was a professor in the University of Bonn since 1921. In 1942 he com- mitted suicide together with his wife and her sister because deportation to a concentration camp was im- minent. Therefore, it is especially moving for us that Shiryaev and your academy gave the seven original letters of Hausdorff to Alexandroff to Bonn Univer- sity. This generous gift was presented to us in Bonn on February 16, 1996 by Professor Shiryaev with Pro- fessors Mishenko and Maltsev present.
At the end of my talk a few words on the correspon- dence between Alexandroff and Hausdorff. The no- tion of topological space defined by neighbourhood axioms goes back to Hausdorff's book "Grundzüge der Mengenlehre" (1914) which had an enormous in- fl.uence on Alexandroff and Urysohn. The first letter of Alexandroff and Urysohn of April 18, 1923 begins as follows: "Hochgeehrter Herr Professor! Schon seit recht langer Zeit streben wir danach, Ihnen die Er- gebnisse, die wir in der von Ihnen geschaffenen Theo- rie der topalogischen Räume erhalten haben, mitzu- teilen". Then some results on compact, locally com- pact and on metric spaces follow. But they indi- cate that they are unable to present everything in a letter. They want to visit Hausdorff in Bonn.
In a later letter they mention Urysohn's metrisa- tion results. The visit of Hausdorff and Urysohn in Bonn takes place in July 1924. Then Alexan- droff and U rysohn travel to the French Atlantic coast and from there write to Hausdorff to thank him and to communicate Urysohn's construction of a seper- able metric universal space, a space in which every
·seperable metric space can be isometrically embed- ded. On August 17, 1924 the catastrophy happens:
Urysohn drowns in the Atlantic. The following letters in both directions center around this terrible acci- dent. Later Alexandroffwrites to Hausdorff about his efforts to edit Urysohn's papers. In 1925 Alexandroff visits Hausdorff in Bonn again. The following corre- spondence concerns the second edition of Hausdorff's book, Urysohn's dimension theory and many other topics. For example, on July 10, 1929 Alexandroff reports about a journey on the Volga with Kolmogo- roff, about Fräulein Noether's very successful visit of Moscow, filling the gaps the Moscow mathematicians had in the field of Algebra, and he reports about new excellent results of Ljusternik, Schnirelmann, Pon- tryagin and Frankl. Hausdorff confesses that through Alexandroff's work he appreciated combinatorial ( al- gebraic) topology for the first time. Originally he had thought that this theory was infinitely dull. Now he loved it, corrected mistakes in the papers of experts and lectured in the Summer term of 1933 in Bonn
"Einführung in die kombinatorische Topologie" . The manuscript is preserved. It was one of his last courses
Hans-Dietrich Gronau
before the Nazis cancelled his professorship.
Bonn mathematicians and several from outside Bonn are planning to edit Hausdorff's collected pa- pers. Brieskorn is writing a biography of Hausdorff.
Brieskorn and Walter Purkert are the driving force of the Hausdorff edition. Purkert wrote an essay about the correspondence Alexandroff-Hausdorff which I have used here. A translation into Russian of the whole correspondence is in preparation.
References
[A] P. Alexandroff, Heinz Hopf zum Gedenken, I. Einige Erinnerungen an Heinz Hopf, Jahresber. Deutsch. Math.- Verein. 78 (1976), 113-146.
[A-H] P. Alexandroff and H. Hopf, Topologie I, Springer- Verlag (1935).
Adresse des Autors:
Prof. Dr. F. Hirzebruch
Max-Planck-Institut für Mathematik Gottfried-Claren-Str. 26
53225 Bonn
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38. Internationale Mathematik-Olympiade (IMO) Mar del Plata, Argentinien, 1997
von Hans-Dietrich Gronau
Die 38. Internationale Mathematik-Olympiade fand vom 18.-31. Juli in Mar del Plata in Argentinien statt.
Mit 82 teilnehmenden Ländern wurde ein neuer Teilnehmerrekord erzielt, was vor allem durch die Teilnahme von zahlreichen Ländern aus Lateinamerika erreicht wurde.
Auswahl und Vorbereitung der deut- schen Mannschaft
Die Auswahl und Vorbereitung der deutschen Mann- schaft verlief nach bewährtem Verfahren der Vorjah- re. Ca. 130 Schüler qualifizierten sich durch die er- folgreiche Teilnahme an der 2. Runde des Bundes- wettbewerbes Mathematik oder an der Deutschland- Olympiade, der 4. Stufe der Mathematik-Olympiade, für 2 Auswahlklausuren, die Anfang Dezember 1996 geschrieben wurden. Die 16 erfolgreichsten Klau- surteilnehmer bildeten den Kandidatenkreis für die deutsche Mannschaft. Für diese gab es 4 Wochen- endseminare und die traditionelle Abschlußwoche in Oberwolfach. Während dieser Zeit wurden insgesamt 6 Klausuren für alle Kandidaten und eine weitere
Stichklausur für einige Schüler geschrieben. Die 6 Be- sten qualifizierten sich für die !MO-Mannschaft.
Der Wettbewerb
Die internationale Jury, bestehend aus den 82 Dele- gationsleitern und einem Chairman des veranstalten- den Landes, entschied sich nach langen Diskussionen schließlich für 6 Aufgaben für die beiden Klausuren, die einerseits eine gute Mischung nach Schwierigkeits- grad und mathematischen Gebieten sein sollen, ande- rerseits aber auch möglichst keine ,Standard'-Lösun- gen zulassen.
Die Olympiade wird als eine mittelschwere mit guten Aufgaben und guter Punktverteilung in die Geschich- te eingehen. So wurden durchschnittlich 16.1 Punkte
1@1973 Kurt Schwitters, Das gesamte literarische Werk, DUMONT Buchverlag Köln
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