Singularities the Brieskom

Progress in Mathematics Volume 162

Series Editors H. Bass J. Oesterle A. Weinstein Singularities

The Brieskom Anniversary Volume

Y.I.Arnold G.-M. Greuel J.H.M. Steenbrink Editors

Springer Base1 AG Editors: V.I.Amold G.-M. GreueI Department of Geometry and Topology Fachbereich Mathematik Steklov Mathematical Institute Universitiit Kaiserslautem 8, Gubkina Stree D-67653 Kaiserslautem 117966 Moscow GSP-I Germany Russia and

CEREMADE J.H.M. Steenbrink Universite Paris-Dauphine Subfaculteit Wiskunde Place du Marechal de Lattre de Tassigny Katholieke Universiteit Nijmegen Postfach 3049 Toemooiveld F-75775 Paris Cedex 16e NL-6525 ED Nijmegen France The Netherlands

1991 Mathematics Subject Classification 14B05, 32SXX, 58C27

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

Deutsche Bibliothek Cataloging-in-Publication Data

Singularities : the Brieskom anniversary volume / V. 1. Amold ... ed. (Progress in mathematics ; VoI. 162) ISBN 978-3-0348-9767-9 ISBN 978-3-0348-8770-0 (eBook) DOI 10.1007/978-3-0348-8770-0

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ISBN 978-3-0348-9767-9

987654321 Dedicated to Egbert Brieskorn on the Occasion of His 60 th Birthday Prof. Dr. Egbert Brieskorn Contents

Preface ...... Xlii Gert-Martin Greuel Aspects of Brieskorn's mathematical work XV

Publication list ...... XXlll

Chapter 1: Classification and Invariants

Yuri A. Drozd and Gert-Martin Greuel On Schappert's Characterization of Strictly Unimodal Plane Curve Singularities ...... 3 Introduction ...... 3 1 Preliminaries...... 4 2 Main theorem ...... 6 3 Ideals of ideal-unimodal plane curve singularities...... 17 References ...... 26

Gert-Martin Greuel and Gerhard Pfister Geometric Quotients of Unipotent Group Actions II 27 Introduction ...... 27 1 Special representations ...... 29 2 Free actions ...... 32 References ...... 36

Helmut A. Hamm Hodge Numbers for Isolated Singularities of Non-degenerate Complete Intersections ...... 37 Introduction ...... 37 1 Mixed Hodge numbers for the link and the vanishing cohomology ...... 38 2 Nondegenerate complete intersections ...... 48 References ...... 59

vii viii Contents

Weiming Huang and Joseph Lipman Differential Invariants of Embeddings of Manifolds in Complex Spaces ...... 61 1 Normal cones...... 63 2 Specialization to the normal cone ...... 67 3 Differential functoriality of the specialization over lR ...... 70 4 Multiplicities of components of C(V, W) ...... 73 5 Relative complexification of the normal cone ...... 78 6 Segre classes ...... 87 References ...... 92

Andras Nemethi On the Spectrum of Curve Singularities 93 1 Introduction...... 93 2 The properties P(N) (2::::: N ::::: (0)...... 94 3 Positive results. The case of plane curve singularities ...... 96 4 The arithmetical approach revisited. Dedekind sums ...... 98 5 Germs without property P(N)...... 100 References ...... 101

Mihai Tibiir Embedding Nonisolated Singularities into Isolated Singularities 103 1 Introduction...... 103 2 The main construction...... 104 3 Homotopy type of the Milnor fibre ...... 108 4 Zeta-function of the monodromy ...... 111 References ...... 114

Chapter 2: Deformation Theory

Andrew A. du Plessis and Charles T. C. Wall Discriminants and Vector Fields...... 119 1 Introduction...... 120 2 General theory of the discriminant ...... 123 3 Construction of discriminant matrices and vector fields...... 126 4 Vector fields on the target of stable maps ...... 127 5 Vector fields in the source ...... 129 6 The relation between vector fields in source and target ...... 134 7 The instability locus and the discriminant matrix ...... 135 References ...... 138 Contents ix

Wolfgang Ebeling and Sabir M. Gusein-Zade Suspensions of Fat Points and Their Intersection Forms 141 Introduction ...... 141 1 p-fold suspensions of icis ...... 142 2 Convenient equations and the corresponding real picture ...... 143 3 A distinguished set of vanishing cycles for the icis {x + zP = 0, x ± y2 = O} ...... 145 4 Enumeration of vanishing cycles and the definition of their orientations ...... 146 5 The equivariant intersection form. The equivariant Picard-Lefschetz formula for the Ap - 1 singularity ...... 150 6 The intersection form of the suspension ...... 152 7 Sketch of the proof of Theorem 2 ...... 155 8 Relations between vanishing cycles of the distinguished set ...... 158 9 Examples...... 159 References ...... 165

Claus H ertling Brieskorn Lattices and Torelli Type Theorems for Cubics in lP'3 and for Brieskorn-Pham Singularities with Coprime Exponents ...... 167 1 Introduction...... 167 2 Hypersurface singularities and polarized mixed Hodge structures 171 3 Brieskorn lattice ...... 175 4 The invariant BL ...... 180 5 Semiquasihomogeneous singularities with weights (~, ~, ~, ~) ... 183 6 Brieskorn-Pham singularities with pairwise coprime exponents .. 188 References ...... 193

Eugenii Shustin Equiclassical Deformation of Plane Algebraic Curves ...... 195 1 Introduction...... 195 2 Preliminaries...... 198 3 Proof of Theorem 1.1 ...... 199 4 Proof of Corollary 1.3 ...... 202 5 Proof of Theorem 1.4 ...... 203 References ...... 203

Victor A. Vassiliev Monodromy of Complete Intersections and Surface Potentials ...... 205 1 Introduction...... 205 2 Vanishing homology and local monodromy of complete intersections ...... 207 3 Surface potentials and Newton-Ivory-Arnold theorem...... 211 x Contents

4 Monodromy group responsible for the ramification of potentials ...... 214 5 Description of the small monodromy group and finiteness theorems in the cases n = 2 and d = 2 ...... 222 6 Proof of Theorems 7, 8 ...... 226 References ...... 233 Appendix to the paper of V.A. Vassiliev (by Wolfgang Ebeling) ..... 235 References ...... 237

Chapter 3: Resolution

Klaus Altmann P-Resolutions of Cyclic Quotients from the Toric Viewpoint 241 1 Introduction...... 241 2 Cyclic quotient singularities ...... 242 3 The maximal resolution ...... 244 4 P-resolutions...... 246 References ...... 249

Antonio Campillo and Gerard Gonzalez-Sprinberg On Characteristic Cones, Clusters and Chains of Infinitely Near Points ...... 251 1 Introduction...... 251 2 Characteristic cones, complete ideals and sandwiched varieties ...... 252 3 Cones and constellations of infinitely near points ...... 254 4 Clusters and chains of infinitely near points ...... 258 References ...... 261

Heiko Cassens and Peter Slodowy On Kleinian Singularities and Quivers 263 Introduction ...... 263 1 Reminder on Kleinian singularities ...... 264 2 McKay's observation ...... 266 3 Symplectic geometry and momentum maps ...... 267 4 Kronheimer's work ...... 271 5 Quivers ...... 273 6 Linear modifications ...... 277 7 Simultaneous resolution ...... 280 References ...... 285 Contents xi

Herwig Hauser Seventeen Obstacles for Resolution of Singularities 289 Introduction ...... 289 1 Basics...... 293 2 Examples...... 295 References ...... 306

Chapter 4: Applications

Enrique Artal-Bartolo, Pierrette Cassou-Nogues and Alexandru Dimca Sur la topologie des polynomes complexes ...... 317 1 Introduction...... 317 2 Demonstration des Theoremes 1 et 2 ...... 321 3 L'inegalite de Kaliman et calcul de l'operateur de monodromie To ...... 324 4 Demonstration du Theoreme 3 ...... 334 References ...... 341

Alan H. Durfee Five Definitions of Critical Point at Infinity ...... 345 1 Introduction...... 345 2 Numerical Invariants ...... 348 3 Resolutions...... 352 4 The Gradient ...... 355 References ...... 358

Joel Feldman, Horst Knorrer, Robert Sinclair and Eugene Trubowitz Evaluation of Fermion Loops by Iterated Residues...... 361 1 Introduction...... 361 2 The integral of a particular rational function over a disk ...... 363 3 The evaluation of Fermion loops ...... 370 Appendix...... 396 References ...... 398

Victor Goryunov and Clare Baines Mobius and Odd Real Trigonometric M-Functions 399 Introduction ...... 399 1 Mobius polynomials ...... 400 2 Odd trigonometric polynomials ...... 404 References ...... 407 xii Contents

Mutsuo Oka Moduli Space of Smooth Affine Curves of a Given Genus with one Place at Infinity ...... 409 1 Introduction...... 409 2 Moduli of polynomials with a fixed irreducible singularity ...... 410 3 Correspondence to affine curves with one place at infinity ...... 413 4 Aut(C2 ) and its action on the space of polynomials ...... 416 5 Structure of the quotient POL(T) (g)/G ...... 422 6 Examples...... 430 7 Appendix. Proof of Lemma 4.6 ...... 431 References ...... 433

Michael Polyak Shadows of Legendrian Links and J+ -Theory of Curves ...... 435 1 Introduction...... 435 2 Arnold's invariants J± of curves and fronts ...... 437 3 J± invariants of I-component fronts ...... 440 4 J-type invariants of multi-component fronts...... 444 5 Shadows and Legendrian shadows ...... 447 6 J-invariants and shadow links...... 452 7 Quantum J+ -type invariants of fronts...... 454 References ...... 457 Preface

In July 1996, a conference in honour of the 60th birthday of Egbert Brieskorn was held at the Mathematische Forschungsinstitut Oberwolfach. It was organ• ised by Gert-Martin Greuel, Vladimir Arnol'd and Joseph Steenbrink. Most people invited to the conference had been influenced in one way or another by Brieskorn's work in singularity theory. We were particularly happy to meet so many people from the Arnol'd school at the conference; it was the first time we were able to enjoy their contribution to this extent. This volume contains papers on singularity theory and its applications, and almost all of them have been written by participants of the conference. In many cases, they are extended versions of the talks given at the conference. The diversity of subjects of the contributions reflects the positioning of singularity theory as a discipline between topology, analysis and geometry, combining ideas and techniques from all of these fields, as well as the broadness of interest of Brieskorn himself. We have arranged the papers according to four main aspects of singularity theory: deformations, classification and invariants, resolution and applications. The latter category contains papers on such diverse topics as affine hypersur• faces, trigonometric functions, Fermion loops and knot theory. Apart from the regular schedule of talks on singularity theory, one day of the conference consisted of lectures devoted to several aspects of the jubilant's mathematical career, and finished with a musical serenade. This part of the program has not been incorporated in this volume, the only exception being the first paper, "Some aspects of Brieskorn's mathematical work" by G.-M. Greuel. All papers published here are original contributions, which will not be published elsewhere. All of them have been carefully refereed.

xiii Some Aspects of Brieskorn's Mathematical Work

Gert-Martin Greuel Universitat Kaiserslautern Fachbereich Mathematik Erwin-Schrodinger-Str. 67663 Kaiserslautern GERMANY e-mail: [email protected]

Lieber Egbert, dear colleagues and friends! I am very happy that this conference on singularities at this wonderful Mathe• matisches Forschungsinstitut Oberwolfach can take place on the occasion of Egbert Brieskorn's 60th birthday, which was a little bit more than a week ago. There have been several conferences on singularities in Oberwolfach - but this is certainly a special one. In this conference we have a special day, the "Brieskorn-day" today, and I am especially happy that this day became possible. As you can clearly see from the programme, the today's speakers are Brieskorn's teacher Prof. Hirzebruch, four of Brieskorn's students and, of course, Prof. Brieskorn himself. I am particularly grateful to Prof. Hirzebruch that when I asked him whether he could give a talk at this occasion, he did not hesitate but im• mediately said yes. He will give us many personal and exciting details of the wonderful discovery of the relation between exotic spheres and singularities. I should also like to thank very much Heidrun Brieskorn, Matthias Kreck and Joseph Steenbrink for preparing a music programme for tonight. They immediately started to exercise when they arrived (even before!). I am sure that we shall have some wonderful music tonight. Thank you very much! Last but not least I want to thank one person especially for his partici• pation at this conference: it is Brieskorn himself. Actually, you can believe me that it was not a trivial task to convince him to come. As many of you know, Brieskorn does not like ceremonies like this, in particular if they concern his own person. Probably he already thinks I should stop talking now. In some sense I would like to agree. On the other hand, I am convinced that Egbert Brieskorn

xv xvi G.-M. Greuel deserves this special day in honour of the person and of his mathematical work which was important

• for his students • for the unfolding of singularity theory, and • for mathematics as a whole.

Before I start to talk about some aspects of Brieskorn's mathematical work, let me mention that Brieskorn is a person whose interest, knowledge and activities reach far beyond mathematics. He loves music and, by the way, knows a lot about the theory of music. He is definitely a very political person with strong opinions. He was actively engaged in the peace movement and is still engaged in projects for saving the environment. As you know, during the past years, he has hecome a semipro• fessional historian in connection with the life and the work of Felix Hausdorff. Actually, he is the editor of the book "Felix Hausdorff zum Gediichtnis I". And - who is surprised - there will soon be a second volume with Brieskorn's biography of Felix Hausdorff. The later work of Brieskorn, with important historical and philosophical contributions, as well as his textbooks is, however, not subject of this short overview. Now let me start with a very short review of part of Brieskorn's mathe• matical work. As you will see, the talks of today are all related to some mathematical theory which nowadays is a grown-up theory, but where, sometime at the beginning, there was a discovery of Brieskorn or a development of a germ of a theory by Brieskorn. Moreover, as I shall try to explain, in all of Brieskorn's work you see the idea of unity of mathematics. Brieskorn's work is led by the idea to combine different mathematical structures, different mathematical categories.

Historically speaking the following different structures are involved. (I hope this will not be too schematic but casts some light on his work):

differential analytic (exotic spheres) resolution deformation (simultaneous resolutions of ADE sin• gularities) Lie groups equations (construction of singularities from the corresponding simple Lie groups) transcendental algebraic (construction of the local GauB-Ma• nin-connection) continuous discrete (generalized Braid groups, Milnor lat• tices and Dynkin diagrams) Some Aspects of Brieskorn 's Mathematical Work xvii

It is quite interesting to notice that perhaps in almost all cases these different structures correspond to the two parts of our brain, as Arnol'd explained in his talk yesterday. Already in his first paper, which, as far as I know, emerged from his dissertation and which has the title "Ein Satz iiber die komplexen Quadriken", Math. Annalen 155 (1964), he proves: Let X be a complex n-dimensional Kahler manifold diffeomorphic to Qn (n• dimensional projective quadric), then

(i) n odd =?- X is biholomorphic to Qn

(ii) n even, n -=I- 2 =?- CI (X) = ±ng (H2(X, Z) ~ Zg, g positive) and if CI (X) = ng, then Xis biholomorphic to Qn- (If n = 2 then X = IP'I X IP'I has infinitely many different analytic structures, the ~2m of Hirzebruch) This was an exact analogue of a previous theorem of Hirzebruch and Kodaira about the complex projective space. (An earlier announcement of the result appeared 1961 in the Notices of the AMS.) The next paper ""iTher holomorphe IP'n-Biindel iiber IP'I", Math. Ann. 157 (1965) treats the same question and gives a complete answer (including the Hirzebruch ~-surfaces ) . His next paper was "Examples of singular normal complex spaces which are topological manifolds", Proc. Nat. Acad. Sci. 55 (1966). This paper contains already the Brieskorn singularity X : Z5 + zi + ... + z; = 0 . He proves: if n :2: 4, n odd, then X is a topological manifold. The result at that time was a big surprise, since in 1961 Mumford had published in his well-known paper "The topology of normal singularities of an algebraic surface and a criterion for simplicity", Publ. Math. IRES 8 (1961), that such phenomena are not possible for surfaces. Now, I should like to switch to the paper "Beispiele zur Differentialtopologie von Singularitaten", Inven• tiones Math. 2 (1966). The results of this paper were a sensation in the mathematical world. xviii G.-M. Greuel

Brieskorn showed that the just discovered exotic spheres (by Kervaire and Milnor) appear as neighbourhood boundaries of singularities and, therefore, can be described by real algebraic equations! As an example, I should like to mention that

{xik - 1 + x~ + x~ + x~ + x~ = O} n 59, k = 1, ... ,28, represent all 28 different differentiable structures on the topological 7-sphere. As far as I understood, the story of discovery of this result was also very exciting and Hirzebruch, who was himself involved in this discovery, will tell us very interesting details. The next two papers "Uber die Auflosung gewisser Singularitiiten von holomorphen Ab• bildungen", Math. Ann. 166 (1966), and "Die Auflosung der rationalen Singularitiiten holomorpher Abbil• dungen", Math. Ann. 178 (1968), contain a proof of the fact that the rational double points admit a simultaneous resolution (after base change), i.e., let X, 5 be smooth, dim X = 3, dim 5 = 1 and f : X ----7 5 be a morphism such that Sing(f) = {x} and (Xf(x),x) is a RDP, then there exists a diagram

X' ~ X f' 1 1 f T ----7 5 'P where X', T are smooth, Sing (f') = 0, 'P is a smooth covering of 5 with 'P-1(f(x)) = {t}, 'I/J is proper, surjective and 'l/Jix; is a resolution of the singu• larities of Xf(x). Moreover, he describes all simultaneous resolutions in terms of invariants of the group G, defining the quotient singularity of type A k , Dk or E 6 , E7 , Eg. These two papers and the next one had an enormous influence on the defor• mation theory of 2-dimensional singularities as well as on the minimal model programme for 3-folds. The paper "Rationale Singularitiiten komplexer Fliichen", Inv. Math. 4 (1968), contains a description of the resolution of quotient singularities and a proof that Some Aspects of Brieskorn 's Mathematical Work xix is the only 2-dimensional factorial analytic ring which is not regular. The ratio• nal surface singularities and, in particular, Brieskorn's work about these play an important role in the subsequent deformation theory of surface singulari• ties. I just mention Riemenschneider, Wahl and later, in connection with the minimal model programme of Mori, Kollar, Reid and others. One of Brieskorn's shortest papers is certainly one of his most important ones: "Singular elements of semi-simple Algebraic Groups", Intern. Congress Math. (1970). In this famous paper Brieskorn shows how to construct the singularity of type ADE directly from the simple complex Lie group of the same type. Moreover, he constructs the whole semi-universal deformation. At the end of that paper Brieskorn says:

"Thus we see that there is a relation between exotic spheres, the icosahedron and E 8 . " which expresses explicitly Brieskorn's idea of the unity of mathematics. But he continues: "But I still do not understand why the regular polyhedra come in." I think that even today there is some mystery in these connections of such different parts of mathematics. Peter Slodowy, who himself developed the theory of singularities and al• gebraic groups further, will give us a talk about this fascinating subject. In 1970 Brieskorn published

"Die Monodromie der isolierten Singularitaten von Hyperflachen" , Manuscripta Math. 2 (1970). In this paper he constructed the local GauE-Manin connection of an isolated hypersurface singularity. This construction gave an algebraic method to compute the characteris• tic polynomial of the monodromy, and in this way combined topological and algebraic structures. In his paper Brieskorn proves that the eigenvalues of mono• dromy are roots of unity by a really very beautiful argument using the solution to Hilbert's 7th problem. This was the time when I was a student in Gottingen, and it was my task to generalize his paper to complete intersections in my Diplomarbeit and later in my dissertation. Much later the work of Brieskorn was taken up by Scherk and Steenbrink and especially by Morihiko Saito who made a tremendous machinery out of it. Also Claus Herling continued Brieskorn's work and applied it to obtain theorems of Torelli type for singularities. He will explain to us the magic Brieskorn lattice H~. xx G.-M. Greuel

Now, let me mention Brieskorn's work about "continuous versus discrete struc• tures" which concerns (generalized) braid groups and actions of these. In his paper "Die Fundamentalgruppe des Raumes der reguHiren Orbits einer endlichen komplexen Spiegelungsgruppe", Inventiones Math. 12 (1971), Brieskorn shows that the fundamental group of EregjW of regular orbits of a complex reflection group W has a presentation with generators gs, s E S, and relations

where both sides have mst factors and where (mst) is a Coxeter matrix. These groups are generalized braid groups and were called Artin-groups by Brieskorn and Saito in "Artin-Gruppen und Coxeter-Gruppen", Inventiones Math. 17 (1972). In that paper these groups were studied from a combinatorial point of view and the authors solved the word problem and the conjugation problem. The connection to singularity theory comes from the fact that for W of type An, Dn , E 6 , E 7 , Es the space EregjW is the complement of the discriminant of the semi-universal deformation of a simple singularity of the same type. This follows from Brieskorn's work "Singular elements of semi-simple algebraic groups" at the International Congress in Nice. Now, the classical braid group of n strings Bn acts on the set of "distin• guished bases" of the Milnor lattice. In his later work Brieskorn and several of his students worked on this subject and Brieskorn expresses at several places that the understanding of this action should be essential for understanding the geometry of the versal unfolding. The first step is to understand the deformation relations between singu• larities within a fixed modality class. The classification of isolated hypersurface singularities with respect to their modality by V.l. Arnol'd is certainly one of the most important achieve• ments of singularity theory. The adjacencies (deformation relations) between these singularities are important as well and still the subject ofresearch articles. In the paper "Die Hierarchie der I-modularen Singularitiiten", Manuscripta Math. 27 (1979), Brieskorn gives all possible deformation relations among Arnold's list of 1- modular (unimodal) singularities. The knowledge of all deformation relations Some Aspects of Brieskorn 's Mathematical Work xxi is important by itself but is particularly interesting because of the different other characterizations of this class of singularities. The deformation relations among the unimodular singularities were re• lated by Brieskorn to a theory which seems to be really far away from sin• gularity theory, the theory of partial compactifications of bounded symmetric domains. This is done in the survey article

"The unfolding of exceptional singularities", Nova Acta Leopoldina 52 (1981).

In the introduction, Brieskorn describes the fascinating relation between these apparently unconnected theories:

"On the one hand we have the deformation theory of the singula• rities in the boundary layer. It has three strata - corresponding to the simply elliptic singularities, the cusps and the exceptional sin• gularities. And it has three stems, corresponding to the tetrahedron, the octahedron, and the icosahedron. On the other hand, we have the three quadratic forms Ek -1 U -1 U obtained from the three ex• ceptional forms E 6 , E 7 , Es by adding two hyperbolic planes. These three forms correspond to the three stems. To each of the forms is associated a bounded symmetric domain D of type IV and two un• bounded realizations belonging to the 0- and I-dimensional boundary components Fo and H of D. Corresponding to these there are cano• nically defined arithmetic quotients D If, D I Zr( Fo) and D I Zr(H) and their partial Baily-Borel compactifications identify with the de• formation spaces associated to the singularities in the three strata: exceptional singularities, cusps, and simply elliptic singularities. "

Still investigating the unimodal singularities which constitute, after the zero• modal (or simple or ADE) singularities, the next class in Arnold's hierarchy of singularities, Brieskorn gives a very fine and detailed study of the Milnor lattice of the 14 exceptional unimodal singularities in

"Die Milnorgitter der exzeptionellen unimodularen Singularita• ten", Bonner Math. Schriften 150 (1983).

The Milnor lattice is the integral middle homology together with the quadratic intersection form with respect to a distinguished basis. It is an arithmetic coding of (part of) the geometry of the versal unfolding and it is a great challenge to see to what extent it reflects the essential features of this geometry. This problem which is embedded in a whole programme is again considered in the paper

"Milnor lattices and Dynkin diagrams", Proc. of Symp. in Pure Math. 40 (1983). XXII G.-lVI. Greuel

Brieskorn poses the question "To which extent is this subtle geometry (the geometry of the unfold• ing of a singularity) reflected in the invariants associated to these singularities? "

In the words of Arnol'd, this programme is the attempt to build a bridge between the two parts of our brain. By the work of Gusein-Zade and Ebeling we understand a lot more, but I guess we are still far away from a complete understanding of the relation between the continuous and the discrete structure of a singularity. Wolfgang Ebeling will talk on these themes. The last paper I should like to mention is "Automorphic Sets and Braids and Singularities", Contemporary Mathematics 78 (1988).

In this paper Brieskorn gives a survey on the action of the braid group on the set of distinguished bases of an isolated singularity. Moreover, he introduces the general concept of an automorphic set ~, which unifies many investigations about the action of the braid group. His statement in the introduction of that paper "The beauty of braids is that they make ties between so many dif• ferent parts of mathematics, combinatorial theory, number theory, group theory, algebra, topology, geometry and analysis, and, last but not least, singularities." shows very clearly Brieskorn's strong belief in the unity of mathematics. Now I come to the end of my talk. I hope I could explain some aspects of Brieskorn's mathematical work and point out that the idea or the wish to show the unity of mathematics, apart from its different realizations, was perhaps one of the leading principles of Brieskorn's work. I should like to finish with a citation. In January 1992 there was a special colloquium in honour of Felix Hausdorff in Bonn. Brieskorn started his talk by citing Hausdorff. Hausdorff had spoken the following words at the grave of the mathematician Eduard Study and had cited Friedrich Nietsche from "Zaratustra" with the words.

"Trachte ich denn nach dem GlUck? Ich trachte nach meinem Werke."

(Do I aim for happiness? I do aim for my work.) Brieskorn said, that this was certainly Hausdorff's leading principle and I should like to add, Brieskorn's too. Publication List

Prof. Dr. Egbert Brieskorn

1. Ein Satz iiber die komplexen Quadriken. Math. Ann. 155, 184-193 (1964).

2. Uber holomorphe W'n-Biindel iiber W'I. Math. Ann. 157, 343-357 (1965).

3. Uber die Auflosung gewisser Singularitiiten von holomorphen Abbildun• gen. Math. Ann. 166, 76-102 (1966).

4. Examples of singular normal complex spaces which are topological mani• folds. Proc. Nat. Acad. Sci. 55, 1395-1397 (1966).

5. Beispiele zur Differentialtopologie von Singularitiiten. Invent. Math. 2, 1-14 (1966).

6. Rationale Singularitiiten komplexer Fliichen. Invent. Math. 4, 336-358 (1968).

7. Die A uflosung der rationalen Singularitiiten holomorpher A bbildungen. Math. Ann. 178,255-270 (1968).

8. Some complex structures on products of homotopy spheres. (With A. Van de Ven). Topology 7, 389-393 (1968).

9. Die Monodromie der isolierten Singularitiiten von Hyperfliichen. Manu• scripta Math. 2, 103-161 (1970).

10. Die Fundamentalgruppe des Raumes der reguliiren Orbits einer endlichen komplexen Spiegelungsgruppe. Invent. Math. 12, 57-61 (1971).

11. Die Monodromie der isolierten Singularitiiten von Hyperfliichen. Mathe• matika, Moskva 15, No.4, 130-160 (1971).

12. Singular elements of semi-simple algebraic groups. Actes Congr. intern. Math., Nice 1970, 2, 279-284 (1971).

13. Sur les groupes de tresses, d'apres V.I. Arnold. Seminaire Bourbaki 1971/72, Expose No. 401. Lecture Notes Math. 317, 21-44 (1973).

14. Artin-Gruppen und Coxeter-Gruppen. (With Kyoji Saito). Invent. Math. 17,245-271 (1972).

xxiii xxiv Prof Dr. Egbert Brieskorn

15. Vue d'ensemble sur les problemes de monodromie. In: Singularites a Cargese, Asterisque 7 et 8, 195-212 (1973). 16. Artin-Gruppen und Coxeter-Gruppen. (With Kyoji Saito). Matematika, Moskva 18, No.6, 56-79 (1974). 17. Sur les groupes de tresses. Matematika, Moskva 18, No.3, 46-59 (1974). 18. Singularities of complete intersections. (With G.-M. Greuel). In: Mani• folds, Proc. Intern. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 123- 129 (1975). 19. Die Fundamentalgruppe des Raumes der reguliiren Orbits einer endlichen komplexen Spiegelungsgruppe. Uspehi mat. Nauk 30, No.6 (186),147-151 (1975). 20. Special singularities - resolution, deformation and monodromy. A series of survey lectures given at the twenty-first Summer Research Institute of the AMS July 29 - August 16, 1974, Humboldt State Univ., Ar• cata/California. Duplicated typoscript, 96 pages. 21. Uber die Dialektik in der Mathematik. In: Mathematiker liber Mathe• matik. (Hrsg. M. Otte). Springer-Verlag, Berlin etc., 221-286, (1974). 22. 0 dia lektice v matematic [-III. (Czech translation). Pokropky Mat. Fyz. & Astr. 24, 33-43, 89-103,163-173 (1979). 23. Singularitiiten. Jber. DMV 78, 93-112 (1976). 24. Die Hierarchie der l-modularen Singularitiiten. Manuscripta Math. 27, 183-219 (1979). 25. The unfolding of exceptional singularities. Nova acta Leopoldina, NF 52, No. 240, 65-93 (1981). 26. Ebene algebraische Kurven. (With H. Knorrer). Birkhauser, Basel, Bos• ton, Stuttgart, XI, 964 pages (1981). 27. Plane algebraic curves. (English translation by John Stillwell). Birk• hauser, Basel, Boston, Stuttgart, VI, 721 pages (1986). 28. Milnor lattices and Dynkin diagrams. Singularities, Summer Inst., Ar• cata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 153-165 (1983). 29. Die Milnorgitter der exzeptionellen unimodularen Singularitiiten. Bonner Math. Schr. 150, 225 pages (1983). 30. Lineare Algebra und analytische Geometrie I. Noten zu einer Vorlesung mit historischen Anmerkungen von Erhard Scholz. Vieweg & Sohn, Braunschweig, Wiesbaden, VIII, 636 pages (1983). Publication List xxv

31. Lineare Algebra und analytische Geometrie II. Noten zu einer Vorlesung mit historischen Anmerkungen von Erhard Scholz. Vieweg & Sohn, Braunschweig, Wiesbaden, XIV, 534 pages (1985).

32. Automorphic sets and braids and singularities. Braids, AMS-IMS-SIAM Jt. Summer Res. Conf., Santa Cruz/Calif. 1986, Contemp. Math. 78, 45- 115 (1988).

33. Brieskorn, Egbert (ed.), Felix Hausdorff zum Gediichtnis. Band I: As• pekte seines Werkes. (In memoriam Felix Hausdorff. Vol. I: Aspects of his work). Vieweg, Wiesbaden, 286 pages (1986).

34. Gustav Landauer und der Mathematiker Felix Hausdorff. In: Gustav Lan• dauer im Gespriich. Symposium zum 125. Geburtstag. Hrsg. Hanna Delf, Gert Mattenklott. Conditio Judaica, Band 18. Max Niemeyer Verlag, Tiibingen, 105-128(1997).

35. Gibt es eine Wiedergeburt der Qualitiit in der Mathematik? To appear in: E. Neuenschwander (ed.): Wissenschaft zwischen Qualitas und Quantitas. Birkhiiuser, Basel, 138 pages (ca. 1998).

36. Felix Hausdorff zum Gediichtnis. Band II: Elemente einer Biographie. (In memoriam Felix Hausdorff. Vol. II: Elements of a biography.) To appear in Vieweg, Wiesbaden, ca. 550 pages (ca. 1998).