93 Bericht aus Bonn

"When I came to Bonn in 1957, Professor Peschl suggested that I ask the Ministry to provide financial support for small conferences. An opportune moment to make such a request occurred when I was offered a chair elsewhere. 'Selbstverst~indlich, Herr Professor' was the reply, and the money was there. I wanted to hold a meeting each year a meeting involving a minimum of organization and preparation. This led to the idea not to fix a program in advance, but to let the participants determine it themselves during the meeting. The method proved very successful: the newest results could always be presented, with mathematicians lecturing on their own work or reporting results of others. The Arbeitstagung quickly became a favorite. Participants return to Bonn year after year and new ones appeared. It has really never become boring and, altogether, each meeting has been a success. "Since 1957 the Arbeitstagung has been held each year except in 1968, in 1973 when it made room for the Hodge Colloquium in Cambridge, and in 1976 when it was replaced by the International Conference on Modular Forms in Bonn. The sum

1979 - 1956 -

3

20 shows that this year's is the twentieth Arbeitstagung." F. Hirzebruch (from the preface to [A]) For the occasion (June 6 - 1 2 , 1979) Bonn commemorated past Arbeitstagungen with a preprint [A] containing the programs and telling photographs. Hirzebruch's pre140 X

X

12C X

tO0 80 o 60 0

9

40

9

20

9 B

X It

0

57 58 59 60 S t G2 63 ~4 65 66 67 68 59 70 7 i 72 73 74 75 75 77 78 79

Number of participants (x = exact number, 9= lower bound). Participants from Bonn are not counted

face was actually in German, "since so many of the participants have by now been to Bonn so often that they ought to understand it." Inevitably the need for funds grew. The Ministry's initial contribution of 1,000 DM increased to 6,000 DM, but has been constant now for many years. This was supplemented by the DAAD (German academic exchange service) and other organizations until the establishment in 1969 of SFB 40, the Sonderforschungsbereich ~r Theoretische Mathematik, Bonn's mathematical research center. The SFB now provides most of the financial support for the meeting. The best witness to the success of the Arbeitstagungen is the collection of programs in the preprint (a pirate edition of which has apparently already appeared in the United States). Here is a brief selection with emphasis on results presented in the earlier years where it is already clear what significance they have had on recent mathematical developments. In 1957 Grothendieck presented his algebraic proof of the generalization of Riemann-Roch-Hirzebruch in a series of lectures "Koh~irente Garben und verallgemeinerte Riemann-Roch-Hirzebruch Formel auf algebraischen Mannigfaltigkeiten" [B-S]. 1959, Grauert explained "How to blow down" in one of a number of talks by Behnke students on analytic spaces and complex manifolds. Grauert's theorem: All the direct images of a coherent analytic sheaf over an analytic space X by a proper holomorphic mapping are coherent [G-R], [G 60], [G 62]. In 1960, Smale and Stallings lectured on smooth and combinatorial homotopy spheres and the Poincar6 conjecture in dimensions t> 5 [Sm], [St], that every combinatorial manifold of the same homotopy type as the n-sphere is homeomorphic to an n-sphere. In a field day for differential topology, Kervaire constructed a manifold which does not admit any differentiable structure [K]. Having already used scissor and paste arguments the previous year, Milnor turned to surgery with a lecture on spherical modifications [M 61a], and in 1961 Stallings presented a 5-dimensional counterexample to the Hauptvermutung in combinatorial topology. Milnor had disproved the Hauptvermutung in dimensions/> 6 [M 61b]. In 1962 the complex analysts were back in force with lectures by Grauert, Remmert, and Stein. Atiyah appears twice in the program, once to talk on harmonic spinors. His second title: "Explanations of my preceding lecture".

94

Grauert, Stein, Remmert (1960)*

A lifting: Atiyah and Grothendieck (1961)

Atiyah and Tits on the Dampferfahrt (1965)

* For different legend see: Swiss Chocolate Factory, Boston, News & Blues # 1.

Bott's lecture for students (1969)

95 Hironaka, the first (recorded) Japanese speaker at an Arbeitstagung, lectured on the resolution of singularities

for algebraic varieties: for any irreducible variety V find a nonsingular projective variety V' birationally equivalent to IT. For characteristic zero Zariski did this for dimensions ~< 3 and Hironaka, in 1964, published his proof for varieties of any dimension [H]. From 1963 on, there was growing emphasis on K-theory with Atiyah on elliptic operators [A-Si 63], [A-Si 68], Hirzebruch reporting an elementary proof of the Bott periodicity theorem [A-Bo 64] and Atiyah again on boundary value problems [A-Bo]. For the following years K-theory continued to be an important feature with applications to homotopy theory (Adams), elliptic operators and index theory (Atiyah, Palais, Segal) [A-Se], [P], and RiemannRoch (Grothendieck), whereas Kuiper contracted the unitary group of Hilbert space. In 1966 Hirzebruch reported on Brieskorn's work in a talk "Exotic spheres and singularities" [Br]. Brieskorn's idea had established an important and surprising connection between algebraic geometry and differential topology which also motivated Milnor's results on singularities of complex hypersurfaces on which he lectured the same year [M 68]. In 1967 many Russians were able to attend for the first time - Anosov, Manin, Postnikov, Shafarevich, Venkov, and the tide started to turn more towards algebraic geometry, algebraic groups, and number theory. 1969: Griffiths, "Algebraic cycles", Schmid "Langland's conjecture" [Sch], Deligne "Hodge theory of singular varieties" [D 70]. After the work of Kirby and Siebenmann on the annulus conjecture and the obstructions for triangulating manifolds had already been discussed, Siebenmann arrived late, expinned his work and that of Hsiang that played a role in these developments, and presented "the most elementary disproof,' of the Hauptvermutung he knew [S]. 1970: Lang "Transcendental mappings", Springer "Discrete series of finite Chevalley groups", Verdier "Serre's duality theorem". The following year Deligne lectured on the Weil conjecture for surfaces of degree 4 inP3 [D 72]. He presented his proof of the Weil conjecture in 1973 at the Hodge colloquium [D 74]. Though no one would claim that there was a "fair" balance of topics at the Arbeitstagungen (the aim after all was not to be fair but to hear something good), they are never excessively one-sided. The trend toward number theory continued, with a liberal smattering of complex analysis, topology, differential geometry, dynamical systems and more. Among the highlights in more recent years were Kiehl on Grauert's theorem, Singer on the ~?-invariant and its relation to real quadratic fields, an application of the index theorem for odd-dimensional manifolds to number theory, Lusztig's lectures in 1974 and 1975 on discrete series representations. Motivated partly by a growing inter-

est in modular forms, the 1976 Arbeitstagung was replaced by the International Conference on Modular Forms [MF]. In 1977 Yang-Mills theory made its Bonn debut (Atiyah), followed up a year later by Atiyah lecturing on instantons and algebraic geometry and Bourgignon on the differential geometry of the Yang-Mills equations. Further attractions in 1978 included Yau's work and solution of the Calabi conjecture and Banchoff's films, "The fourth dimension and computer animated geometry" (from which Figures 1 - 3 are taken). A regular feature of each program is the Festlegung der niichsten Vortrdge, the determination of who will speak on the next two days. The event is not unlike an auction. The lecture room is full. Names and topics are called out from the benches, some prearranged, others not, as is apparent from the possible speakers' occasionally somewhat worried glances. Discussions ensue and likely suggestions emerge. Hirzebruch, standing at the front, sees which faces flinch, nod, smile, or grimace at which suggestions. Within a couple of minutes clear decisions have been reached which, in other circumstances, would take six months of refereeing. The discussions are usually still in full swing as it gradually emerges from Hirzebruch's notes on the blackboard which of the suggested speakers will in fact deliver lectures. The above sketch of impressive mathematical activity from previous programs is a tribute to Hirzebruch and the telling faces he watches most among the participants. Of course, in retrospect the programs look very attractive and one is tempted to forget that the talks were mostly on a very high level which sometimes made them hard

Figure 1. A hyperbolic paraboloid, w = z 2 with domain the lower half of the Riemann sphere and graph (x, y, x 2 - y2, 2xy) in 4-space is projected into the (x, y, u) space. (Banchoff, Strauss)

96 Programm der Mathematischen Arbeitstagung 1979

Mittwoch, den 6. 6.:

J. Tits: On Leech's lattice and sporadic groups

Donnerstag, den 7. 6.: F. Adams: G. Segal's Burnside ring conjecture F. Bogomolov: Converse Galois problems for some Chevalley groups Wang Yuan: Goldbach problem

Freitag, den 8. 6.:

Festlegung der nachsten Vortr~ige D. Vogan: Size of representations L. Berard-Bergery: A new example of Einstein manifolds Ausflug nach Oberwesel. Abfahrt 12.30 Uhr mit Bussen an der Beringstr. 1 nach Koblenz. Abfahrt ca. 13.30 mit Motorschiff ,,Carmen Silva" ab Koblenz ca. 13.30 Uhr.

Samstag, den 9. 6.:

V. Kac: Infinite dimensional Lie algebras G. Mostow: New negatively curved surfaces G. Lusztig: Representations of Hecke algebras

Sonntag, den 10. 6.:

Festlegung der restlichen Vortr~ige B. Gross: Conjectures of Stark and Tate Wu-chung Hsiang: Topological space form problems M.-F. Vigneras: Isospectral but not isometric Riemannian surfaces E. Looijenga: Singularities and generalised root systems

Montag, den 11. 6.:

Parshin: Zeta functions and K-theory Min-Oo: Curvature deformations relating to the Yang-Mills fields G. Harder: Cohomology and values of L-functions

Dienstag, den 12. 6.:

A. Todorov: Moduli of K~hlerian K3-surfaces R. P. Langlands: On orbital integrals for real groups J.-P. Serre: The monster game

97 warned to l o o k for their children playing in the courtyards o f the castle. The loop-holes in the walls are very l o w . " The lectures are also in English which h o w e v e r hardly ever leads to misunderstandings. This year, being special, the b o a t trip was up the R h i n e f r o m K o b l e n z to Oberwesel, past the Loreley. The atmosphere, like every other year, was one o f strawberry cake and mathematics.

References These are n o t i n t e n d e d to be c o m p l e t e (e.g., SGA is missing), but j u s t to refer to publications o f results m a n y o f which were r e p o r t e d in earlier stages at Arbeitstagungen. [AI

[A-Bo]

[A-Bo 64] [A-Se] Figure 2. The Gauss mapping of a biparabolic surface [A-Si 631

[A-Si 681

[B-SI [Brl [D-70I

ID 72] [D 741 [G601

[G621

[G-RI Figure 3. Steiner's Roman surface tetrahedral symmetry

(x/~xz, x/r2yz,x/~xy) with

[HI

[K] to follow; participating in an Arbeitstagung is likely to be strenous. Also in every program is the Dampferfahrt. The preparations behind the scenes are substantial. Participants receive leaflets (in English) w i t h such e x h o r t a t i o n s as "Please wear good shoes. Parents o f small children are

[K-SI ILl

Arbeitstagungen des Math. Inst. und des Sonderforschungsbereichs "Theoretische Mathematik" der Univ, Bonn, SFB 40, Bonn 1979 M. F. Atiyah and R. Bott, The index problem for manifolds with boundary, Bombay Colloquium on Differential Analysis, Oxford Univ. Press (1964), 175-186 M. F. Atiyah and R. Bott, On the periodicity theorem for complex vector bundles, Acta Math. 112 (1964) M. F. Atiyah and G. B. Segal, Seminar on equivariant K-theory, Lecture notes, Oxford Univ., 1965 M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422-433 M. F. Atiyah and I. M. Singer, The index of elliptic operators I, II, III, Ann. of Math. (2) 87 (1968), 484-604 A. Borel and J.-P. Serre, La th~or~me de RiemannRoch, Bull. Soc. Math. France 86 (1958), 97-136 E. Brieskorn, Beispiele zur Differentialtopologie, Inventiones Math. 2 (1966), 1 - 1 4 P. Deligne, Th6orie de Hodge: I, Actes Congs. Intern. Math., 1970, Nice, Gauthier-Villars; II, III, Publ. Math. IHES, no. 40, 1971, p. 5 - 5 7 ; no. 44, 1974 P. Deligne, La conjecture de Weil pour les surfaces K3, Inventiones Math. 75 (1972), 206-226 P. Deligne, La conjecture de Weil, Publ. Math. IHES 43 (1974), 273-307 H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulr/iume komplexer Strukturen, Publ. Math. IHES, no. 5, 1960 H. Grauert, ~rber Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331-368 H. Grauert and R. Remmert, Komplexe R~iume, Math. Ann. 136 (1958), 245-318 H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. (2) 79 (1964), 109-326 M. A. Kervaire, A manifold which does not admit any differentiable structure, Comment, Math. Helv. 34 (1960), 257-270 R. Kirby and L. Siebenmann, Bull. Amer. Math. Soc. 75 (t969) R. P. Langlands, Dimension of spaces of automorphic forms, Amer. Math. Soc. Proc. Symposia in Pure Math., vol. 9, 1966, p. 253-257

98 [M 61a]

[M 61b]

[M 681 [MF1 [P] IS] [Schl

[Sm]

IStl

J. W. Milnor, A procedure for killing the homotopy groups of differentiable manifolds, Amer. Math. Soc. Proc. Symp. Pure Math. III (1961), 39-55 J. W. Milnor, Two complexes which are homeomorphic but combinatorially distinct. Ann. of Math. 74 (1961), 575-590 J. W. Milnor, Singular points of complex hypersurfaces, Ann. Math. Studies, Princeton Univ. Press, 1968 Modular Functions of one Variable V, VI, Bonn 1976, LNM 601,627, Springer-Verlag 1977 R. Palais, Seminar on the Atiyah-Singer index theorem, Ann. Math. Studies, Princeton Univ. Press, 1965 L. C. Siebenmann, Topological maiaifolds, Actes Congr6s Int. Math. 1970, Tome 2, 133-163 W. Schmid, On a conjecture of Langlands, Ann. of Math. (2) 93 (1971), 1-42. W. Schmid, On the realization of the discrete series of a semisimple Lie group, Rice University Studies 56 (1970), 99-108 S. Smale, Generalized Poincar6's conjecture in dimensions greater than four. Ann. of Math. (2) 74 (1961), 391-406 J. R. Stallings, Polyhedral homotopy-spheres, Bull. Amer. Math. Soc. 66 (1960), 485-488

N o w available

H. Hasse

Number

Theory

English Translation Edited and Prepared for Publication by H.G. Zimmer 1980.49 figures. XVII, 638 pages (Grundlehren der mathematischen Wissenschaften, Band 229) Cloth DM 9 8 , - ; approx. US $57.90 ISBN 3-540-08275-1 Distribution rights for the Socialist Countries: Akademie-Verlag, Berlin Contents: The Foundations of Arithmetic in the Rational Number Field.- The Theory of Valued Fields.- The Foundations of Arithmetic in Algebraic Number Fields. Hasse's classic work, originally published in 1949 and then in a second, thoroughly revised edition in 1962, is now available in Englisch, revised once more. The main topic of the book is the foundations of number theory in algebraic number fields and algebraic function fields in one indeterminate. Hasse's approach derives from the works of Kronecker and Kummer, and of his own teacher Hensel, whose valuation theory plays a major role in the book. Traditionally this treatment stands in contrast to the ideal-theoretic approach historically associated with Dedekind, Hilbert, and Emmy Noether. The publication of this English edition coincides with the growing interest in the divisor-theoretic methods of Kronecker and Kummer, exemplified by the recent publication (also by SpringerVerlag) of Kummer's collected works. Hasse's book is still up-to-date and remains the only comprehensive presentation of this approach. Only the most important results are presented in Theorem-Proof style, and much space is devoted to giving insight into the structure of the~ subjects discussed by expounding on them from many points of view. ~.

Springer-Verlag Berlin Heidelberg NewYork

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