Preface Issue 2-2015

Jahresber Dtsch Math-Ver (2015) 117:91–92 DOI 10.1365/s13291-015-0116-z PREFACE

Hans-Christoph Grunau1

Friedrich Hirzebruch, who died in 2012 at the age of 84, was one of the most important German mathematicians of the second half of the twentieth century. This applies not only to his mathematical achievements—mainly in topology, geometry and number theory—and his huge school of at least 52 Ph.D. students. He also made enormous efforts and had great success in reshaping mathematics in Germany after the Second World War. The Sonderforschungsbereich Theoretische Mathematik, the Arbeitstagung and the Max-Planck-Institut für Mathematik played an important role in making Germany once again a place which mathematicians from all over the world would like to visit. The Deutsche Mathematiker-Vereinigung (DMV) owes a particularly deep gratitude to Friedrich Hirzebruch who served twice as its chairman, in 1962 and 1990. In 1962, one year after the wall was built, the Gesellschaft für Math- ematik der DDR was founded and urged to branch off from the DMV. One purpose of his second election in 1990 was that he should chair the reuniﬁcation of both societies and to celebrate at the same time the 100th anniversary of the DMV. In the 28 years in between he ensured that the ties to the East were not severed. One of his best known former Ph.D.-students, Don Zagier, gives a fairly detailed picture of all these and many more facets of Friedrich Hirzebruch’s life within mathematics. Often, pictures are available only in a corrupted form and one has to try to reconstruct the original ones. One may think for instance of a real valued function describing the gray-value of the picture at the actual position. If one assumes that the corruption mechanism (the corruption mapping) is known up to some “noise” one has to seek a preimage of the corrupted picture under this mapping. In principle one could apply the corruption mapping to all possible pictures and search for those

B H.-Ch. Grunau [email protected]

1 Institut für Analysis und Numerik, Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg, Germany 92 H.-Ch. Grunau whose images minimise a suitable distance functional from the given corrupted picture. Due to the “bad” properties of the corruption mapping this problem is in this form not well posed. In order to resolve this issue, one adds a regularisation term containing ﬁrst and second order derivatives of the unknown preimage to the distance functional. Gabriele Steidl explains in her survey article on “Combined ﬁrst and second order variational approaches for image processing” the recent development of such techniques, and illustrates them by two rather different applications. Functions of bounded variation and related function spaces are fundamental for Gabriele Steidl’s survey article and are also the topic of a recent book by Jürgen Appell, Józef Banas, and Nelson José Merentes Díaz, which is reviewed by Michel Willem. The second review concerns a book by Luis Barreira and Yakov Pesin on smooth ergodic theory and is provided by Gerhard Knieper. Jahresber Dtsch Math-Ver (2015) 117:93–132 DOI 10.1365/s13291-015-0114-1 HISTORICAL ARTICLE

Life and Work of Friedrich Hirzebruch

Don Zagier1

Published online: 27 May 2015 © Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2015

Abstract Friedrich Hirzebruch, who died in 2012 at the age of 84, was one of the most important German mathematicians of the twentieth century. In this article we try to give a fairly detailed picture of his life and of his many mathematical achievements, as well as of his role in reshaping German mathematics after the Second World War.

Mathematics Subject Classiﬁcation (2010) 01A70 · 01A60 · 11-03 · 14-03 · 19-03 · 33-03 · 55-03 · 57-03

Friedrich Hirzebruch, who passed away on May 27, 2012, at the age of 84, was the outstanding German mathematician of the second half of the twentieth century, not only because of his beautiful and influential discoveries within mathematics itself, but also, and perhaps even more importantly, for his role in reshaping German mathematics and restoring the country’s image after the devastations of the Nazi years. The field of his scientific work can best be summed up as “Topological methods in algebraic geometry,” this being both the title of his now classic book and the aptest description of an activity that ranged from the signature and Hirzebruch-Riemann-Roch theorems to the creation of the modern theory of Hilbert modular varieties. Highlights of his activity as a leader and shaper of mathematics inside and outside Germany include his creation of the Arbeitstagung, his presidency of the Deutsche Mathematiker- Vereinigung during two especially critical periods and his later services to the Euro- pean Mathematical Society and the International Mathematical Union, the founding of the Max Planck Institute for Mathematics in Bonn, and his role in preserving mathematical contacts between the Federal Republic of Germany and the German Demo- cratic Republic and Soviet Union during the Soviet period and later in establishing close mathematical links between Germany and many other countries, notably Japan,

B D. Zagier [email protected]

1 Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany 94 D. Zagier

Fig. 1 F. Hirzebruch with the author at the Max Planck Institute for Mathematics ca. 1985

Poland and Israel. He was a superb lecturer, teacher, and expositor of mathematics and above all a man whose human qualities were an inspiration and a model for those around him. Accounts of several periods of Hirzebruch’s life and activities, told in his inim- itable style, can be found in a number of his own articles, while more systematic accounts are given in the article by Joel Segel, the long video interview with Matthias Kreck for the archives of the Simon Foundation, and the AMS memorial article edited by Michael Atiyah and myself, which also contains portraits of him by several of his friends and associates. The exact references for all of these are given in the short bib- liography at the end of this article. A book-length biography by Winfried Scharlau is also in preparation. The emphasis here is different: I would like to both recount the main stages of Hirzebruch’s career and to give a somewhat more in-depth look at some of his mathematics and at the many ways in which he helped shape the development of mathematics during his time. The article will be organized roughly chronologically, but alternating between the events and activities of his life and a description of his mathematics. This article owes much to Michael Atiyah and Matthias Kreck, with whom I had many helpful discussions. Above all I would like to thank my wife, Silke Wimmer- Zagier, who participated in every step of the writing. Our friendship with Fritz Hirze- bruch transformed both of our lives.

1 Early Years and Education

Friedrich (“Fritz”) Ernst Peter Hirzebruch was born to Dr. Fritz Hirzebruch and Martha Hirzebruch, née Holtschmit, on October 17, 1927, in Hamm, North Rhine- Westphalia. His love of mathematics was awakened early by his father, a mathematics teacher and headmaster of the local secondary school, one of whose students, Karl Stein, was later to become one of Hirzebruch’s mentors and a famous mathematician in his own right. Already as a√ child he was fascinated by concepts like other number systems or the irrationality of 2, and apart from a brief ﬂirtation with astronomy he never seriously imagined pursuing any other profession than that of a mathematician. Life and Work of Friedrich Hirzebruch 95

His youth was marked by the Third Reich and the Second World War. While still a teenager he was drafted into “Deutsches Jungvolk” (German Youth) and assigned to anti-aircraft positions, where he officially scanned the skies for enemy planes but mostly thought about mathematics. He was drafted into the army soon before Ger- many’s collapse, escaping active combat by a stroke of luck, and soon afterwards was captured by Allied forces and put into a POW camp, where he continued to do mathematics, famously writing some of it on toilet paper. At that time, whenever asked about his war experiences, he would answer laconically with the single sentence “In the period from mid-January 1945 to the first of July 1945 I went through fatigue duty, military service, and war captivity.” He was released in July 1945. Hirzebruch began his undergraduate studies in December 1945, soon after his 18th birthday and, of course, also soon after the end of the war. The university of Münster was in ruins and in the first months the unique auditorium available to the Faculty of Philosophy and Science could only be used for mathematics lectures one day every three weeks, but the situation gradually got better, with lectures every two weeks in the following spring and every week by Fall 1946. His teachers at Münster were Heinrich Behnke and the above-mentioned Karl Stein. From them he learned the function theory of several complex variables that was to be his first field of research and the subject of his thesis. During his sixth semester, in 1948, he saw an advertisement for a three-week stint of agricultural work in Switzerland, to be followed by a fourth week in a freely chosen place in Switzerland, and decided to apply. With the support of Behnke and Heinrich Scholz, for whose seminar on logic and foundations he had been one of the student assistants, he was awarded this “prize,” which involved 14 hours of hard physical labor a day for the first three weeks, with a financial remuneration of 40 Swiss francs during the last week. His stay in Switzerland was to determine the course of his future mathematical life.

2 Switzerland, 1948Ð1950

Before Hirzebruch left Münster, Behnke had written to Heinz Hopf in Zürich to ask whether he would receive and advise the young Fritz. The recommendation letter he sent was also memorable because the secretary to whom it was dictated, Ingeborg Spitzley, soon afterwards became a mathematics student and four years later Hirze- bruch’s wife! After completing his stint as a farm laborer, which taxed his weakened constitu- tion to the utmost but which he seems to have rather enjoyed nevertheless, he went to Zürich to visit Hopf and Paul Bernays. He was received with extraordinary friend- liness (this was only three years after the end of the war) by the Hopfs, who put him up in their own house and treated him like a son. From Hopf he had his first introduction into differential topology, with Hopf’s brand-new article “Zur Topologie der komplexen Mannigfaltigkeiten” as the guiding text. Here he learned about almost complex structures and how to show that such manifolds as S4, S8 or the complex projective plane with reversed orientation do not have any and hence in particular cannot be complex manifolds. In [12] he writes that these first discussions were the 96 D. Zagier starting point for both his own first publication, in Crelle’s Journal in 1953, and his later joint paper with Hopf, in the Mathematische Annalen in 1958. After his return to Münster, Hirzebruch continued his studies with Behnke, Stein and Scholz for one more semester, until Spring 1949. With the support of his teachers, a scholarship for him to study at the ETH in Zürich was set up, with a matching fund established that the ETH could use later for a Swiss student to study in Germany (as indeed happened, in 1951). He spent the period from April 1949 till August 1950 at the ETH, receiving their Silver Medal (and a prize of 1000 Swiss francs) for his solution of the prize problem of developing the relation between differential topology and the theory of analytic functions in many variables (“now, who might have suggested that topic?” he asks drily in [12]) in July 1950 and completing his doctoral dissertation at the same time. He had to return to Münster to defend it, in the same month, because his official supervisor was still Heinrich Behnke, but he himself always said that his true doctoral supervisor was Heinz Hopf, who sometimes referred to him as his illegitimate Doktorsohn. The subject of the thesis was the resolution of singularities (or “Hopf σ -process”) on complex algebraic surfaces. Here he found a toroidal construction (as such things later came to be called) and discovered that by iteration of the σ - or blowing-up process one often obtained a graph of rational curves whose self-intersections were given by the partial quotients of a continued fraction, a harbinger of the links to number theory that were to be one of the main characteristics of his later work. In particular, he n = n−q showed that the resolution of the singularity w z1z2 , where 0

In this connection he made the incorrect assertion that the resolution process could obviously never produce a cycle—an error that he noticed soon afterwards and for which he would atone most handsomely many years later (as we will discuss in detail below) with his discovery of the resolution of the cusp singularities of Hilbert modular surfaces by cycles of rational curves related to periodic continued fractions. In this thesis Hirzebruch introduced the family of algebraic surfaces Σn that are now called Hirzebruch surfaces and proved that the manifolds Σn and Σm are homeomorphic whenever n − m is even, but are always distinct as algebraic or complex surfaces. From the modern point of view this is a simple result: The surfaces Σn are the total spaces of certain bundles, so the classification of bundles together with the intersection form on cohomology gives the homeomorphism classification, and the first Chern class distinguishes them in the complex world. But at this time these fundamental concepts were either still unknown or had just been discovered, and Hirze- bruch himself was to become one of the main people to contribute to their study. Hirzebruch returned to Germany in the winter semester of 1950 to take up a position as Assistent in Erlangen, where he stayed for two years and published the results Life and Work of Friedrich Hirzebruch 97 of his thesis and other related work. But it was in Princeton, where he spent the two following years, that his mathematics really took off.

3 The Golden Years: Princeton 1952Ð54

Hirzebruch himself always used the words “golden years” when speaking about the time he spent at the Institute for Advanced Study in Princeton. It was a period of intense excitement for him, both mathematically and personally. He had received an invitation, probably engineered by Heinz Hopf, in Erlangen, and left by boat on August 9, 1952; his bride Inge (Ingeborg, née Spitzley), whom he had married in a civil ceremony just before leaving, joined him in November in Princeton, where they celebrated a church wedding. Hirzebruch’s name was already known to some of his American colleagues before he arrived, and he was met in person by Newton Hawley and Donald Spencer when his ship docked in Hoboken. He began working with them and Kunihiko Kodaira right away, in particular meeting Spencer and Kodaira (who were themselves in the middle of their now-famous collaboration) every day from 9 till 5 at the Institute [6]. He thus got to know “from the horse’s mouth” the newest and most up-to-date advances in algebraic geometry, the theory of sheaves, bundles, and characteristic classes, whose definitive formulation and exposition he was to give in the book Neue topologische Methoden in der algebraischen Geometrie. The institute was full of outstanding personalities, not only the famous scientists who had come from Germany and Eastern Europe before the war like Albert Einstein (whom Inge once famously mistook for the gardener), Kurt Gödel (whom Hirzebruch remembered always seeing immersed in conversation with Einstein), Marston Morse (whose seminars Hirzebruch attended), or Hermann Weyl (who took great interest in Hirzebruch’s work and befriended both him and his wife), but also a whole generation of brilliant young European mathematicians who were to shape the coming decades and many of whom became Hirzebruch’s close friends: Borel, Bott, Atiyah, Lang, Singer, Serre, Milnor, and many others. The mathematics that they discussed and developed during this period were to shape all of Hirzebruch’s future work, as well as much of the mathematics of the second half of the twentieth century, and in the case of two of them, Atiyah and Borel, led to extremely fruitful collaborations that we will discuss below in detail. A high point of Hirzebruch’s two years in Princeton was the birth of his first child, Ulrike, in 1953. (His other two children were born after his return to Germany, Bar- bara in Hamm in 1956 and Michael in Bonn in 1958. Ulrike and Barbara both later studied mathematics and worked in related fields, Ulrike in mathematics publishing and Barbara as a teacher, while Michael became a doctor.) But these years also marked the birth of the two great theorems that bear his name, the Hirzebruch Sig- nature Theorem and the Hirzebruch-Riemann-Roch Theorem. We now describe each of these in turn.

4 Characteristic Classes, Genera, and the Signature Theorem

In topology, as in the rest of mathematics, one wants to find computable invariants that permit one to distinguish non-isomorphic objects, where “non-isomorphic” here 98 D. Zagier might mean “non-homeomorphic” or “non-diffeomorphic” according to the situation. These invariants can be purely numeric, or else algebraic objects like the homology or cohomology of a manifold (each of which is a graded vector space endowed with additional structure, the intersection form or cup product, respectively), or yet more subtle invariants like characteristic classes in the cohomology ring. The most interesting invariants often have more than one description. For instance, the Euler characteristic e(M) of a closed manifold M can be defined purely combinatorially as the alternating sum of the numbers of simplices of various dimensions in a triangulation of M (but then one has to demonstrate that this definition is independent of the triangulation, as was done famously by Euler for the 2-sphere, with Euler number 2), or algebraically as the alternating sum of the dimensions of its homology or cohomology groups, or, in the case when M is a complex n-manifold, as the integral of the nth (or total) Chern class of the tangent bundle of M: e(M) = c(T M) = cn(T M), [M] . M The signature theorem gives a similar description of the signature, which is an important topological invariant of oriented manifolds. (We recall briefly the definition: if M is a closed oriented manifold of even dimension n, then the intersection form on H n/2(M) is non-degenerate by Poincaré duality and is symmetric if n/2 is even, in which case Sign(M) is defined to be the signature of this form, namely, the num- − 2 +···+ 2 − 2 −··· 2 ber r s if the form can be written as x1 xr xr+1 xr+s for some n/2 basis x1,...,xr+s of H (M; R).If4dim M, then Sign(M) = 0.) To explain it, we must first say something about characteristic classes, to whose theory Hirzebruch contributed substantially (in particular through his work with A. Borel, discussed below), and also about multiplicative sequences and genera, which are among the most wonderful of all of his inventions, beautifully combining theoretical and computational ideas. We start with characteristic classes, the two main cases for our discussion being 2i theChernclassesci(E) ∈ H (X) of a complex vector bundle E over a topological 4i space X and the Pontryagin classes pi(E) ∈ H (X) of a real vector bundle E over X. (A third case, the Stiefel-Whitney classes in H ∗(X; Z/2Z) of a real vector bundle over X, will be omitted, although it plays a role in several of Hirzebruch’s works.) The basic one is the Chern class. Recall that a (topological) d-dimensional complex vector bundle E over X is a map p : E → X that “looks locally like a product with Cd ”, in the sense that X can be covered by small open sets U with isomorphisms between p−1(U) and U × Cd that are fixed up to the action of a continuous map from U to GL(d, C). In the case d = 1 (complex line bundles), the bundle E is uniquely 2 determined by a characteristic class c1(E) ∈ H (X; Z), called its first Chern class, and we define the total Chern class of E to be c(E) = 1 + c1(E). We also require the total Chern class of a direct sum of two bundles to be the product of their total Chern classes, so that if a d-dimensional bundle E is a sum of d line bundles L1,...,Ld , with first Chern classes x1,...,xd , then we have d d c(E) = (1 + xα) = ci(E), α=1 i=0 Life and Work of Friedrich Hirzebruch 99

2i with ci(E) ∈ H (X; Z) being the ith elementary symmetric polynomial in the xj .It turns out that these requirements and the functoriality property c(f ∗(E)) = f ∗(c(E)) for the pull-back of a bundle E over X by a continuous map f : Y → X already suffice to define Chern classes in general (by the so-called splitting principle). Fi- nally, Pontryagin classes can be defined in terms of Chern classes: if E is a real vector bundle over X (defined like complex bundles, but with C replaced by R), then E ⊗R C is a complex bundle having (up to 2-torsion) non-zero Chern classes only for even indices, and one defines the individual and total Pontryagin classes = − i ⊗ C ∈ 4i = ∈ 4∗ of E by pi(E) ( 1) c2i(E R ) H (X) and p(E) i pi(E) H (X), respectively. We should emphasize that the algebraic approach described here was developed to a large extent by Hirzebruch and his collaborators (especially Atiyah and Borel), and is more general and easier to work with than the approach based on geometry as used earlier by Pontryagin, Chern and others. We now come to Hirzebruch’s multiplicative sequences. These are multiplicative characteristic classes (i.e., functions assigning to any bundle E over a space X ∗ ∗ ∗ an element χ(E) ∈ H (X; R) and satisfying χ(f E) = f (E) and χ(E1 ⊕ E2) = χ(E1)χ(E2)) that are canonically associated to the Chern or Pontryagin class of E 2 and to a power series f(x)= 1 + a1x + a2x +··· with coefficients in some ring R (forususuallyQ). In the complex case the multiplicative sequence χ is defined by f + formally factorizing the Chern class c(E) as α(1 xα) with each xα of degree 2 and setting χ (E) = f(x ), which makes sense because the product f α α = + + 2 +··· f(xα) 1 a1xα a2xα α α = + + 2 + 2 +··· 1 a1 xα a2 xα a1 xαxβ α α α<β

= + + 2 + 2 − +··· 1 a1c1(E) a2c1(E) a1 2a2 c2(E) can be written completely in terms of the elementary symmetric functions ci(E) of the formal roots xα. (A trivial case is obtained by taking f(x)= 1 + x, in which case χ (E) is simply c(E).) An exactly similar construction works for real bundles f + and Pontryagin classes, where we write p(E) as α(1 yα) with formal roots yα of degree 4 and deﬁne χ (E) ∈ H 4∗(X) for any power series h(y) by χ (E) = h h = ⊗ C α h(yα), or equivalently by χ (E) χ (E R ) where f(x) is the even power h f series h(−x2). In particular, if √ y y y2 2y3 h(y) = √ = 1 + − + +··· tanh y 3 45 945 (corresponding to f(x)= x/tan x), then we get the total Hirzebruch L-class

L(E) = χ√ √ = 1 + L1(E) + L2(E) +··· y/tanh y 4i of the real bundle E, where each Li(E) belongs to H (X; Q), the ﬁrst values being 2 3 p −p + 7p2 2p − 13p1p2 + 62p3 L = 1 , L = 1 , L = 1 ,.... 1 3 2 45 3 945 100 D. Zagier

Hirzebruch’s Signature Theorem states that, for all oriented manifolds M, one has Sign(M) = L(T M), [M] , where TM is the tangent bundle of M. About its proof, he liked to say: “Having conjectured the signature theorem, I went to the library at Fine Hall. Thom’s Comptes Rendus note on cobordism had just appeared. This finished the proof.” This modest statement is nearly true, but only because of a key insight that Hirzebruch had already had, namely, that every multiplicative sequence χ defines a “genus.” To explain this, h we first recall the concept of cobordism. Two closed oriented n-manifolds M1 and M2 are called cobordant if the disjoint union of M1 and M2 (= M2 with reversed orientation) bounds an oriented (n + 1)- manifold.

The equivalence classes under this equivalence relation form an abelian group Ωn, with addition given by disjoint union and negation by change of orientation (which works because M M bounds M ×[0, 1]), and Ω∗ := Ωn forms a graded ring via [M1][M2]=[M1 × M2] (Cartesian product). It is relatively easy to show that the signature defines a homomorphism from this ring to Z, i.e., that it behaves additively and multiplicatively under disjoint union and Cartesian product and that it vanishes for orientable boundaries. Such a map is called a genus. On the other hand, if h is any power series with rational coefficients and χ the corresponding multiplicative h sequence, then the function χh(M) := χ (M), [M] is also a genus, the additivity h and multiplicativity being easy and the cobordism invariance a consequence of the fact that the boundary of an oriented manifold is null-homologous on this manifold (or simply of Stokes’s theorem if one thinks of cohomology classes as represented by differential forms). This realization had led Hirzebruch to conjecture the signature theorem, and Thom’s result saying that Ω∗ ⊗ Q is the free polynomial algebra on the complex projective spaces CP2n (n = 1, 2,...) indeed quickly finished the proof, because it reduced the statement of the theorem for arbitrary orientable manifolds to the statement for these projective spaces, where it is easy: the cohomology of CP2n is 2n+1 2 2n the truncated polynomial algebra Z[x]/(x ) with x ∈ H and x , [CP2n] = 1, its signature is always 1 (because H 2n = Z · xn and the self-intersection of xn is 1), and the Chern class of its tangent bundle is (1 + x)2n+1,so + x 2n 1 L T(CP n) , [CP n] = , [CP n] 2 2 tanh x 2 + x 2n 1 = Coefficient of x2n in tanh x dx = Resx= 0 (tanh x)2n+1 Life and Work of Friedrich Hirzebruch 101

d arctanh t = Res = t 0 t2n+1

= Coefﬁcient of t2n in arctanh (t) = 1. An exactly similar calculation leads to the following more general result.

Theorem Every genus χ : Ω∗ →√Q is equal to χh for a unique power series h(y) ∈ y ∞ 2n+1 1 + yQ[[y]], given by h(y) = √ where g(t) = χ(CP ) t . g−1( y) n=0 2n 2n+1

The signature theorem immediately had many applications. Rokhlin and Schwarz, and independently Thom, used it in the 50’s to prove that Pontryagin numbers of manifolds, i.e., the evaluations on their fundamental classes of arbitrary polynomials in the Pontryagin classes of their tangent bundles, are combinatorial invariants; it was shown later by Novikov that they are in fact topological invariants. Even more spectacular was the theorem of Milnor in 1956 that the topological 7-sphere has more than one differentiable structure. (The possibility of the existence of exotic differentiable structures had already been pointed out by Hirzebruch in 1953.) In fact there are precisely 28 such structures, as Kervaire and Milnor showed later, where the factor 7 in the number “28” comes directly from the coefﬁcient of p2 in the Hirzebruch L- polynomial L2 and from the signature theorem, applied to an appropriate 8-manifold constructed from the 7-sphere in question.

5 The Hirzebruch-Riemann-Roch Theorem and Its Applications

We discussed Hirzebruch’s signature theorem in considerable detail because it was so close to his heart and is so characteristic of his mathematics. His Riemann-Roch theorem is even more important, but since it involves the same main ingredients, characteristic classes and multiplicative sequences, we can be briefer here. We begin by recalling the notion of the genus of a compact Riemann surface, which has both a topological and an analytic version. The topological one, going back essentially to Riemann’s original paper of 1857, is a topological invariant that can be defined for any closed oriented 2-dimensional manifold (as half of the first Betti number) and is known to determine this manifold up to homeomorphism. The analytic version, also going back to Riemann, applies only when the surface has a complex structure (so is a 1-dimensional complex projective variety, or complex curve) and involves this analytic structure rather than merely the underlying topology: the genus equals the dimension of the space of holomorphic 1-forms on the curve. Its significance is shown by Riemann’s inequality, which says that for any divisor D = nixi of degree deg(D) = ni on the curve, the dimension l(D) of the vector space of meromorphic functions whose divisor (= set of zeroes and poles, counted with multiplicity) is ≥−D satisfies l(D) ≥ deg(D) + 1 − g, where g is the genus. This inequality was completed by Roch to the famous Riemann-Roch formula l(D) − l(K − D) = deg(D) + 1 − g, where K is the canonical divisor. 102 D. Zagier

In the later 19th century, these results were generalized to surfaces, in particular by Castelnuovo and Max Noether. In the first decades of the twentieth century, algebraic geometers studied possible ways to define the genus for an arbitrary n- dimensional projective smooth algebraic variety X. These included the geometric genus gn(X) (where gi(X) denotes the dimension of the vector space of holomorphic i-forms on X, so that g1(X) is the usual genus when n = 1), the arithmetic genus = n − i χ(X) i=0( 1) gi(X), and two further numbers denoted by pa(X) and Pa(X) whose definitions we omit. It was conjectured by Severi in 1950, and proved later by n Kodaira and Spencer using sheaf theory, that pa(X) = Pa(X) = (−1) (χ(X) − 1). A fourth definition was proposed by John Todd in 1937 as a certain combination of the Chern numbers of X and was conjectured by him to be equal to the arithmetic genus. But it was neither known how to prove this equality nor how to describe the required combination of Chern numbers in general. The solution to both problems that Hirzebruch found used the multiplicative sequences that we have already seen. He defined the Todd class as the multiplicative characteristic class χ corresponding f = − −x = + 1 + 1 2 +··· to the power series f(x) x/(1 e ) 1 2 x 12 x , so that the first components of the Todd class Td(E) = 1 + T1 +··· of a complex vector bundle E are given by 2 c c + c2 c c T = 1 ,T= 1 ,T= 1 2 , 1 2 2 12 3 24 4 2 2 −c + 4c c2 + c1c3 + 3c − c4 T = 1 1 2 , 4 720 where ci denotes the ith Chern class of E. Just as before, this also defines a genus, by associating to any smooth complex variety X the value of the (top-dimensional component of) the Todd class Td(X) of its tangent bundle evaluated on its fundamental class. The result that Hirzebruch proved for projective algebraic varieties X, generalizing formulas that had been found in low-dimensional cases by Todd, was that this Todd genus coincides with the arithmetic genus: χ(X)= Td(X), [X] = Td(X). X = = + 1 = 1 [ ] = For n 1 this gives Td(X) 1 2 c1(X) and hence χ(X) 2 c1(X), X 1 = − 2 e(X) 1 g. Thus in this case the formula reduces to the previously discussed equality of the topological and analytic definitions of the genus of a Riemann surface. To generalize the full Riemann-Roch equality, which involves a divisor (or equivalently, complex line bundle) on a Riemann surface as well as just the surface itself, we need to consider not only an n-dimensional complex variety X, but also a (holomorphic) complex vector bundle E on X of some dimension d and to find a formula for the holomorphic Euler characteristic of its sheaf cohomology groups, i.e., for the number n χ(X,E)= (−1)i dim H i(X, E). i=0 If X is a curve (Riemann surface) and E the line bundle corresponding to a divisor D on X, then this sum reduces to the expression l(D) − l(K − D) appearing in the Life and Work of Friedrich Hirzebruch 103 classical Riemann-Roch formula. The Hirzebruch-Riemann-Roch theorem gives the answer in the general case: χ(X,E)= ch(E)Td(X). X − Here ch(E) is the Chern character of E, defined as e yβ where formally c(E) = β d + β=1(1 yβ ). Hirzebruch’s proof of his Riemann-Roch theorem was a characteristic combination of high-powered topological machinery and virtuoso computation. He wrote a long letter to Todd explaining his results and received a reply saying “Both Hodge (to whom I have showed your letter) and myself are very interested in this work . . . I like your general expression. Incidentally, I have had to revise a long-held opinion that the Princeton School of Mathematicians despises anything in the nature of algorithmic ingenuity.” The theorem was a major achievement, both in its own right and as a pre- cursor of Grothendieck’s generalization (first presented by him at the Mathematische Arbeitstagung of 1957, of which more later) and above all of the Atiyah-Singer index theorem, which is one of the key mathematical discoveries of the twentieth century and of which both the signature theorem and the HRR theorem are special cases. At a more mundane level, the HRR theorem is a powerful tool to solve concrete problems in the interface between topology and complex algebraic geometry. Hirze- bruch loved questions of this sort and in 1954 published a collection of 34 “problems on differentiable and complex manifolds,” many of which continued to be studied for decades. As a typical example, Problem 14 asked for the possible Chern classes of all almost complex structures on Pn(C). This was solved some 40 years later by his student Michael Puschnigg. For instance, for n = 4 there are (up to complex conjugation) precisely three almost complex structures, with Todd genus 0, 1 and 1001!

6 Münster, the “Topologische Methoden,” and the Work with Armand Borel

During Hirzebruch’s two years at the Institute of Advanced Study he had been on leave from the University of Erlangen. On his return to Germany, he spent the year 1954–55 in Münster with a fellowship from the state of North Rhine-Westphalia, but then went back to Princeton the following year—this time not at the IAS but as an assistant professor at the university. During his postdoctoral year in Münster he wrote his landmark book Neue topologische Methoden in der Algebraischen Geometrie and submitted it to the university as his “Habilitationsschrift” or habilitation thesis, which was a necessary preliminary for eligibility for a professorship. This book gave not only a description of the work he had done in Princeton, culminating in the Hirzebruch-Riemann-Roch theorem, but also a beautifully written exposition of the foundations of the whole ﬁeld: the theory of bundles, sheaves, cohomology, characteristic classes, and all of the surrounding machinery. He had originally intended to publish it as a journal article, but in the end it became a book in the Springer “Ergebnisse der Mathematik” series. This book immediately gave him an international reputation, and was the “Bible” of the ﬁeld for decades. 104 D. Zagier

These two intermediate years between his IAS stay and his professorship in Bonn were also the period when he completed and wrote up his huge joint work with Borel. Technically speaking, their collaboration consisted of only a single paper, but it was a mammoth paper that was published under a single title “Characteristic classes and homogeneous spaces” in three parts in the same journal (American Journal of Mathe- matics 1958, 1959 and 1960). In this very influential paper the authors gave a systematic development of characteristic classes, based on a fusion of ideas from algebraic topology and homotopy theory with a deep analysis of algebraic groups and their associated Lie algebras and root systems, that provided the foundations for the whole field as well as for the Riemann-Roch theorem and for much of Hirzebruch’s other work. The contents are best summarized by the opening paragraph of the paper: It is known that the characteristic classes of a real or complex vector bundle may be interpreted as elementary symmetric functions of certain variables, which are 1, 2 or 4 dimensional cohomology classes. If we consider the tangent bundle of the coset space G/U of a compact connected Lie group modulo a closed subgroup, it turns out that these variables may be identified with certain roots of G (or their squares). Our first purpose is to establish this connection between roots and characteristic classes, which is the basis of this paper, and to compute the characteristic classes of certain well-known homogeneous spaces. These results are then applied in particular to G/T (T maximal torus of G), and to other algebraic homogeneous spaces, where they lead to relations between characteristic classes, Betti numbers, the Riemann-Roch theorem and representation theory; they are also used to discuss multiplicative properties of the Todd genus and other genera in fibre bundles with G/U as fibre. As an application, we get a divisibility property of the Chern class of a complex vector bundle over an even dimensional sphere which yields some information about certain homotopy groups of Lie groups. Hirzebruch presented the joint work with Borel in a series of lectures at the sym- posium “Algebraic Topology and its Applications” in Mexico City in August 1956. Here he also presented his “proportionality theorem” and its applications to the theory of automorphic forms, which he published separately in the conference proceedings. The proportionality theorem says that the Chern classes of the compact quotient of a bounded symmetric domain X by a properly discontinuous and fixed-point free group action are proportional to the Chern classes of the compact dual X , a compact algebraic manifold into which X is naturally embedded (which for the simplest case, the upper half-plane, is just the Riemann sphere). In the introduction to his paper, Hirzebruch says that a special case of the theorem can already be found “between the lines” in a paper by Igusa and that the proof he gives, simpler than his original one, is based on a suggestion by Borel.

7 Herr Professor Hirzebruch

Already at the end of his year in Münster, Hirzebruch had been invited to give a lecture at Bonn University at which to his surprise he saw many physicists and biolo- gists. In fact, he was being “inspected” and soon afterwards received an offer of a full Life and Work of Friedrich Hirzebruch 105 professorship in Bonn. (As he learned later, faculty opinions on this appointment had been divided, with some complaining that he was much too young, but the statement by one of the professors that this was a failing that time would surely cure had finally won the day.) It was not at all clear whether he should accept this appointment at a university that at the time had no great reputation in mathematics, and he was urged to stay in Princeton by many people—including Emil Artin, who however later changed his mind and himself returned to Germany. That he decided not to make his career in America was due partly to Inge’s preference for living in Germany and bringing up the children there, but surely also partly because he already had the vision of helping to restore German mathematics and the credibility of Germany as a decent country. Despite many offers of professorships at other universities, Hirzebruch never left Bonn, remaining there uninterruptedly (apart from a sabbatical year in Princeton in 1959–60 and many short stays at universities around the world) from 1956 until his retirement in 1993 and maintaining a full undergraduate teaching load even after founding the Max Planck Institute and directing it from 1980 onwards. During his years at the university, and largely under his influence, the number of full professors in mathematics increased from three to six, and he attracted such stars of pure mathematics as Jacques Tits (in 1964), Wilhelm Klingenberg (1966), Günter Harder (1969) and Egbert Brieskorn (1975). He served as dean of the Faculty of Science from 1962 to 1964, as well as taking on many other tasks in the academic community, e.g. as a principal editor of both Topology and Mathematische Annalen from 1961 until his retirement and as a very active referee for both the Deutsche Forschungsgemeinschaft (German Science Foundation) and the Alexander von Humboldt Stiftung. Hirzebruch was an extraordinary teacher. He loved teaching at all levels and made a point of “accompanying” the same undergraduates over several years, teaching the beginning courses one year, then the second-year courses, and so on. His lectures seemed very relaxed and even leisurely, always finishing exactly on time and with the available blackboards exactly and very legibly filled. But when I once had to sub- stitute for him during a two-week absence, and despite having his perfectly prepared notes, I had to race so hard that all the students complained about my speed, yet man- aged to cover only two-thirds of the material that he would have presented. The same, of course, held for his advanced lectures, in which he would seem to be introducing his subjects at such a gentle pace and in such an elementary way that his hearers often felt that they could have digested more, only to find when they tried to go through the material in detail that they had been given enough mathematical nourishment to serve them for weeks. But, as Atiyah, describing this knack, once memorably said, “rabbits do not appear out of hats unless they are put there!” An anecdote that he liked to recount perhaps explains why or how he developed this trick of deceptive simplicity. When he was still a very young professor, he was invited to give a colloquium lecture at another German university but was warned that the level there was not very high and that he should at least start at an elementary level. So, although his goal was to give his audience an inkling of his higher- dimensional generalization of the Riemann-Roch theorem, he spent most of the hour explaining the case of curves, which had been known since the 19th century. To make doubly sure, he even spent most of his time on the case of the projective line, for which the theorem is basically empty, reducing essentially to the statement that a rational function of one variable has equally many zeros and poles. And, not to take any 106 D. Zagier risks at all, he even recalled the definition of the projective line as the union of the complex number field and a point at infinity. When the lecture ended, there was dead silence, broken finally by the professor who had invited him and who obviously felt that there had to be at least one question. The question was: “This point at infinity, Herr Professor: does it really exist?” The other part of Hirzebruch’s teaching that must be mentioned was his role as research adviser. He supervised the theses of over fifty doctoral students, including (to mention only some of those with whom I have had personal contact) Klaus Lamotke, Dietmar Arlt, Rabe von Randow, Egbert Brieskorn, Detlef Gromoll, Klaus Jänich, Karl Heinz Mayer, Wolfgang Meyer, Friedhelm Waldhausen, Winfried Scharlau, Di- eter Erle, Helmut Hamm, as well as Werner Meyer, Walter Neumann, Erich Ossa, Matthias Kreck, Ulrich Karras, Alois Scharf and Karl-Heinz Knapp. I mention this last group of seven separately because it was they, together with Ulrich Koschorke (who had done his Ph.D. under Richard Palais) who formed the group of graduate students and postdocs that I became part of when I arrived myself in Bonn in 1970. It was a terrific group, but what impressed me above all was the way they did mathematics, concentrating not only on their own research and on the problems that they could discuss with their supervisor, but also organizing seminars on high-level topics like spectral sequences or Spanier-Whitehead duality that they chose themselves and at which each participant was assigned some part of the theory to study in detail and to present to the others. It was impossible to go through such a training and not become a mathematician!

8 The Arbeitstagung

Already one year after coming to Bonn, Hirzebruch got a sum of 1000 Deutschmarks from the rector of the university to organize a new conference, which he called the Arbeitstagung (literally, “work meeting”). This meeting, which has survived as an annual or bi-annual event until today, was one of his most successful and original inventions. The first Arbeitstagung, which took place in 1957 from Saturday, July 13, until the following Saturday, had only six participants. But what a six! They were Michael Atiyah, Hans Grauert, Alexander Grothendieck, Friedrich Hirzebruch, Nico- laas Kuiper, and Jacques Tits. All of them lectured, but the dominating event was un- questionably Grothendieck’s lecture series “Kohärente Garben und verallgemeinerte Riemann-Roch-Hirzebruch Formel auf algebraischen Mannigfaltigkeiten” in which, in a total of 12 hours of talks, filling all of Saturday afternoon and Monday, Tues- day and Wednesday mornings, Grothendieck presented his generalization of the Hirzebruch-Riemann-Roch theorem. Already at this first AT the principle was established which was to give the meetings their unique flavor: it was decided only in discussions after the participants had come together who would speak, and how often, and about what. One year later the number of speakers had doubled: Abhyankar, Bott, Grauert, Grothendieck, Hirzebruch, Kervaire, Milnor, Puppe, Remmert, Serre, Stein and Thom. Bott lectured about his periodicity theorem, already a year old but not yet Life and Work of Friedrich Hirzebruch 107 known to the participants, and about applications to divisibility properties of Chern numbers, and many of the other talks were on related topics. The beginnings of topological K-theory and of the Atiyah-Hirzebruch collaboration can perhaps be traced here. And in 1962 the big excitement was the talk by Atiyah on “Harmonic spinors” (and its follow-up with the nice title “Explanations of my preceding lecture” four days later), in which the Atiyah-Singer index theorem—still in the process of being worked out in Oxford and still partially conjectural—was introduced to the “general public” for the first time. Hirzebruch gives a very lively account of these first Arbeit- stagungen in [11]. In the subsequent years the Arbeitstagung became more and more popular, with the number of participants reaching well over 250 in some of the later years. The “democratic” principle was maintained, but in a very special form of “guided democ- racy”: except for the opening talk on Friday afternoon, which was decided in advance and was almost always given by Atiyah (with the joke running that he always talked about the index theorem, but never said the same thing twice), the entire program was decided “on the spot” by the participants in three program discussions. These were strategically placed on Friday, Sunday, and Tuesday afternoons, so that no speaker ever had more than one or two days notice that he or she was going to talk (though it could often be observed that participants who had produced a result of note during the preceding 12 months came equipped with extensive notes, just in case). These discussions were occasionally heated, with people shouting out names of possible speakers and topics (the rules, not always respected, required that one had to name both, and that the nominee was not allowed either to refuse or to pick a different subject) and the chairman, who was usually Hirzebruch himself, writing them neatly on the board or, sometimes, skillfully not hearing them. There were also special social events that became part of the tradition: the Rector’s party in the ballroom of the university’s Poppelsdorf Castle, the boat trip on the Rhine (during which half of the participants were typically so engrossed in mathematical discussions that they forgot to look at the sights), followed by a walk up a hill in one of the nearby Rhine villages to get one’s strawberry cake, and, at least in the first decades when the number of people was not too unmanageable, often a party in the Hirzebruchs’ apartment in Endenicher Allee. Not surprisingly, people who had participated once usually wanted to come again, and the list of “regulars” grew from year to year. The Arbeitstagung was always an exciting event, both because of its special organization and the endless curiosity about who would be speaking, but also because of its almost universally high level. It was very often here that one heard about a mathematical breakthrough for the first time or first became aware of a new star. (Among the début appearances that I personally remember were those of Gerd Faltings, Si- mon Donaldson, Frances Kirwan and Maxim Kontsevich.) As the years went by, it lost something of its original innocence and appeal, with too many participants for everyone to get to know everyone else personally and with occasional hard feelings about who had been chosen to speak, something that Hirzebruch deplored. But even so it remained for many decades one of the key events in the annual calendar of countless European and non-European mathematicians (Fig. 2). 108 D. Zagier

Fig. 2 50 years of Arbeitstagung. Left, top: A. Grothendieck, M. Atiyah 1958. Center: R. Bott, M. Atiyah, F. Hirzebruch 1960. Bottom: G. Segal, M. Atiyah, J. Tits 1965. Right, top: R. Thom, F. Hirzebruch 1965. Center: participants of the 30th Arbeitstagung 1991 with S. Kobayashi, R. Kühnau, E. Calabi, F. Hirze- bruch. Bottom: Hirzebruch giving the opening lecture in 2007, 50 years after the ﬁrst Arbeitstagung

9 The Collaboration with Atiyah and the Invention of Topological K-theory

Hirzebruch’s friendship with Michael Atiyah was one of the central axes of his life and of his mathematics. Together with Raoul Bott and Iz Singer they became an inseparable group—later known affectionately as the “Gang of Four”—during Hirze- bruch’s visits to Princeton, and the many discussions of the foursome during this time were important in the development of K-theory by Atiyah and Hirzebruch, of the Atiyah-Singer index theorem, and of the equivariant Atiyah-Bott-Singer index theorem [10]. Atiyah and Hirzebruch met many times in America, England and Germany, and wrote a total of nine papers together. As already mentioned, Atiyah gave the opening lecture, the only one fixed in advance, at nearly every Arbeitstagung. Hirze- Life and Work of Friedrich Hirzebruch 109 bruch proudly recounted that Atiyah had spoken 32 times at the 30 Arbeitstagungen that he organized, from 1957 to 1991, as well as at several of the “Arbeitstagungen of the Second Series” organized by the four directors of the MPIM every two years from 1993 onwards. Their friendship, which included also their families, lasted until Hirzebruch’s death, and joint portraits of the two of them hang in both Bonn and Edinburgh. The main period of their collaboration was from 1959–1962, during which they wrote eight joint papers (in three languages) containing the development of topological K-theory and many applications. This work, which began when they both had a sabbatical term at the IAS in 1959, was inspired on the one hand by the algebraic- geometric K-theory of Grothendieck—itself an outgrowth of his generalization of the Hirzebruch-Riemann-Roch theorem—and on the other by the Bott periodicity theorem. In fact there were two theories, a real and a complex version. Both immediately became powerful tools in topology and index theory, and they were also important as the first widely used examples of generalized cohomology theories. The basic idea of topological K-theory is to associate to a space (finite CW- complex) X a group K(X) in the complex case or KO(X) in the real case, defined as the Grothendieck group of complex (resp. real) bundles over X, i.e., as the group of formal Z-linear combinations of such bundles where [E] is identified with the sum of [E ] and [E ] whenever one has a short exact sequence of bundles 0 → E → E → E → 0 over X. A key ingredient, as already mentioned, is the Bott periodicity theorem, which Bott had just discovered. This theorem describes a periodicity—with period 2 in the complex case and period 8 in the real case—in the homotopy groups of classical groups: πk(U) = πk+2(U) and πk(O) = πk+8(O), where U and O denote the infinite unitary and orthogonal groups, respectively. An- other formulation of this (in the complex case) is that the second loop space Ω2U of U is homotopy equivalent to U, or equivalently Ω2BU ∼ Z × BU, where BU is the classifying space for stable complex vector bundles. The K-group K(X) of a space X can be obtained as the set of homotopy classes of maps from X to Z × BU, and this leads finally to a Z/2Z-graded generalized cohomology theory Kn(X), and similarlytoaZ/8Z-graded generalized cohomology theory KOn(X) in the real case. The realization that these groups of vector bundles constitute generalized cohomology theories, i.e., functorial maps from topological spaces to abelian groups satisfying the Eilenberg-Steenrod axioms, was a key insight that had a profound effect on the algebraic topology of the subsequent decades. Among the important tools developed by Atiyah and Hirzebruch in the course of their study of topological K-theory was their famous spectral sequence, which they introduced for that case but which applies equally to any other generalized coho- ∗ mology theory h .TheE2 term of this spectral sequence is the ordinary cohomology group H i(X, hj (pt)) of the space X with coefficients in the generalized cohomology of a point, and the spectral sequence converges to hi+j (X). Hirzebruch and Atiyah planned to write a book-length exposition of topological K-theory, but in the end never did, though Atiyah later wrote such a book alone based on a course of lectures that he gave at Harvard in 1964. The theory quickly became a basic tool of algebraic topology, with many later important applications, 110 D. Zagier including two very famous ones by J.F. Adams: a simple proof of the Hopf invariant one problem, and an upper bound for the number of linearly independent vector fields on spheres. The starting point of their joint work on K-theory is described by Hirzebruch in his report [11] on the early Arbeitstagungen: “At the ICM 1958 Atiyah mentioned to me that Grothendieck’s method gives the integrality of the Todd genus of a stable almost complex manifold X by embedding X in an even-dimensional sphere. Dis- cussion showed that one can prove this way that all RR-numbers are integers. We were on our way to the topological K-theory and the ‘differentiable Riemann-Roch theorems’.” The paper referred to here is their first joint one, in which a new proof is given that certain divisibility properties for Chern numbers of complex algebraic varieties that followed from the Hirzebruch-Riemann-Roch theorem were in fact true for all almost complex manifolds. This question, raised by Hirzebruch, had already been answered largely by work of Borel-Hirzebruch and Milnor using cobordism theory and the Adams spectral sequence, but the new proofs, based on the emerging K-theory, were completely different and much more natural. Similar ideas led to their second paper (the one in French) giving conditions for the minimum dimension of a sphere into which certain manifolds can be embedded. Several other of their joint papers gave further important applications of K-theory and index theory to concrete problems. One, for instance, showed that the Hodge conjectures are false for integer cohomology (the case of rational cohomology is of course still open), while another was related to Hirzebruch’s earlier discovery of a relation between Steenrod squares and Todd polynomials. Their last joint paper was written in 1970, several years after the others, and combined in a beautiful way Hirzebruch’s theory of genera with Atiyah-Singer index theory. One defines the “A-hat” class as the multiplicative characteristic class associated x/2 = − x2 +···∈Q[[ ]] in the way explained above with the power series sinh(x/2) 1 24 x . Since this is an even power series, the corresponding characteristic class can be written as a polynomial in the Pontryagin rather than Chern classes and hence gives a cohomology class A(T M) ∈ H ∗(M; Q) for any smooth orientable manifold M. Borel and Hirzebruch proved that the corresponding genus A(M) = A(T M), [M] is always integral if M is a spin-manifold (one with vanishing second Stiefel-Whitney class), and Atiyah and Hirzebruch proved that it vanishes if M admits a non-trivial S1-action. The integrality comes from index theory, since from the Atiyah-Singer index theorem one obtains an interpretation of A(M) as the index of the Dirac operator acting on spin-bundles over M. This work later took on a new life in connection with the discovery by Ochanine and others of the elliptic genus, in which Hirzebruch took a lively interest and which will be described later. Although Atiyah and Hirzebruch wrote no further joint papers after 1970, they always remained in close mathematical (as well as of course personal) contact, and a conjecture that Hirzebruch made relating a topological invariant of cusp singularities of higher-dimensional Hilbert modular varieties to special values of certain number- theoretical L-functions was the inspiration for later papers of Atiyah with Patodi and Singer on the index theorem for manifolds with boundary and with Donnelly and Singer to prove Hirzebruch’s conjecture. Life and Work of Friedrich Hirzebruch 111

Fig. 3 F. Hirzebruch, newly appointed honorary fellow of the Royal Society of Edinburgh in 2010, with M. Atiyah

10 Founding an International Center: The First Attempts

Already in 1955 during his second stay in Princeton as assistant professor, Hirze- bruch had met Reimar Lüst (later the third president of the Max Planck Society) and had communicated his enthusiasm for the IAS. He was convinced that this model—a non-university institution that had few permanent members, but invited both recog- nized senior and promising young mathematicians for shorter or longer stays, in a relaxed atmosphere and free of academic duties and worries—was a wonderful way to promote mathematical research, and his goal from then on was to set up an institute of similar type in Germany. The awareness that the impetus for the creation of the IAS had been the exodus of both Jewish and non-Jewish scientists from Germany in the 1930’s, and the conviction that Germany had a debt to repay, certainly also contributed to his determination. As we have already told, Hirzebruch had received the offer of a chair in Bonn shortly before leaving for his second Princeton stay in 1955, and when he and his new young family decided to return to Germany and accept this position, he immediately started working towards his goal. With invitations to Nicolaas Kuiper and Raoul Bott as guest professors and the initiation of the iconic Arbeitstagung, the first small but very important steps were taken. In 1958, the newly founded European Atomic Energy Community (EURATOM) had plans to found a European university-level institution and, within that framework, also a European Institute for Advanced Study for Mathematics (EUROMAT). Each EURATOM country was invited to send two representatives to the first plan- ning meeting, those from Germany being Hellmuth Kneser from Tübingen and Wil- helm Süss, who was then the director of the Mathematics Research Institute in Ober- wolfach. Since Süss was very ill, Hirzebruch was asked to replace him and agreed. The meeting took place and its participants were urged to present the plans to the ministries of their respective countries, which Hirzebruch did, at the same time also working hard to get more government support for Oberwolfach, of which Kneser had assumed the direction after Süss’s death the same year. However, in July of the same year Hirzebruch learned that a French research institute in mathematics and physics, the IHES, had been founded by the industrialist 112 D. Zagier and mathematician Léon Motchane, with himself as first director and with financial contributions from French and Italian companies as well as from EURATOM. It was clear that the EUROMAT proposal was doomed and Hirzebruch immediately gave it up. Instead, still in the same month, he wrote up a proposal for turning Oberwol- fach from a conference center into a full-fledged research institute. The ministry of interior affairs, to whom this proposal was addressed, was very favorable. Further discussions with Kneser and with Paul Hübinger from the ministry followed, during which it emerged that the optimal solution would be the creation of a new Max Planck Institute for Mathematics with two seats: the original one in Oberwolfach, that would serve as a meeting and conference place, and the other one—for practical reasons—in Freiburg, where the guest researchers and their families would live. A new “Society for mathematical research” that would serve as a link between political and research organizations was founded in June 1959 and immediately began discussions with the Max Planck Society, and already in December 1959, during Hirzebruch’s sabbatical leave from Bonn to Princeton, he got a formal letter asking whether he would be willing to conduct the negotiations for the creation of the new institute and to serve as its first director if the application was approved. This letter explicitly mentioned the guiding principle of the MPS that each new institute should be founded around the science and the visions of one person, not just in a chosen field. Despite some reservations, in particular about the impact of the new job on his teaching activities, he agreed, writing that the new institute should have about 15 visiting positions and that the visitors’ fields should vary. On January 6, 1960, the Senate of the Max Planck Society approved the project of setting up an MPIM with Hirzebruch as director, and in May 1960 a major German newspaper reported that Adolf Butenandt, who had just been named the new President of the MPS, had announced the Society’s plan to found two new MPI’s in the humanities (!), of which one was to be in mathematics. But it was not to be. As part of the process of founding the projected institute, the MPS asked for recommendation letters from respected external mathematicians, and two of these reports, from Siegel and from Courant, were negative—for the wrong reasons. Siegel thought that Hirzebruch’s main field, abstract algebraic geometry, was a passing fad and like the rest of modern abstract mathematics had no future. Apart from this very wrong prediction, Siegel seems not to have captured that in any case the future MPIM would aim to pursue fields covering a large part of mathematics, not just the research interests of its director. The objections of Courant, who was an admirer of Hirzebruch, were different: He considered it a crime to rob students of such a splendid lecturer. This, too, was unfounded since Hirzebruch had written clearly in his application of his intention to continue teaching his full load at the university even if the MPIM was founded (and indeed did so when many years later it finally was). But Courant was also an opponent of non-university mathematical institutes in general, and like Siegel disliked the trend to the abstract in mathematics. The two reports, in particular Courant’s, were decisive, and in November 1960 the MPS, very confused by the conflicting advice that they had received and in any case not knowing quite how to handle an institution centered on visitors rather than laboratories, formally turned down the application for the new institute. Life and Work of Friedrich Hirzebruch 113

11 The Sonderforschungsbereich Theoretische Mathematik

But Hirzebruch’s determination to fight for establishing an international research center in Germany continued, and there were new political currents favourable to this idea. In 1969, there was a big initiative from the Deutsche Forschungsgemeinschaft (German Research Foundation) to set up “Sonderforschungsbereiche” (Collaborative Research Centres) within many German universities. These were meant to foster mul- tidisciplinary research around an already existing excellence kernel, beyond the normal university programs or possibilities. With primarily R. Leis and H. Unger from the more applied directions and Hirzebruch for the more theoretical side, the Faculty of Sciences had prepared an application for an SFB, and Hirzebruch was chosen to present this application to the DFG. In fact, he came back with two SFB’s, one each in the more “pure” and more “applied” fields: the SFB 40 “Theoretical Mathematics” under his leadership and the SFB 72 “Approximation and Mathematical Optimiza- tion” under that of Leis and Unger. In the following years, with the help of Silke Suter, who assisted him first part time as a graduate student and then full time, he slowly built up the scientific and administrative structures of the new center and developed the model that would later serve as the starting point for the Max Planck Institute. The SFB 40 bloomed, with the Arbeitstagung and an already extensive guest program as its flagships. Among the early visitors were many eminent mathematicians, including Robert Langlands, Armand Borel, George Mackey, Pierre (Peter) Gabriel and Serge Lang. Many of them also lectured at the university, enriching the students’ program. It was sometimes said at the time that there were many mathematicians in the world who knew only two words of German: “Arbeitstagung” and “Sonderforschungsbereich.” Through the many visitors to the Sonderforschungsbereich, and later also to the Max Planck Institute, Hirzebruch became more and more closely involved with the mathematics and mathematicians of other countries. His work for and with the countries of Eastern Europe, and then with Israel, will be described below in some detail, but here we mention Japan in particular because it was precisely the flexible structure of the visitors’ program of the SFB that permitted this relationship to flourish. Hirzebruch had first visited Japan in 1972, on the invitation of his friend Kodaira, who was then the dean of the University of Tokyo, to give IMU lectures, and had met many of Kodaira’s brilliant students whom he later invited to Bonn. In the following years, the stream of visitors from Japan steadily increased, sometimes so numerous that a whole seminar would be held only in Japanese. There is an old MPI photo that shows a group of more than 20 Japanese who were visiting at the same time. This was due in particular to the fact that in those times Japanese mathematicians could only go abroad for a total of two years in their academic career, and since going abroad with their families meant that they practically moved their whole household, they were happy to get invitations for two years continuously, something that the SFB and later the MPI could do but that few universities or other research institutions could. Because Hirzebruch did so much for the Japanese connection, he later received the prestigious Seki prize from the Japanese Mathematical Society and the Order of the Holy Treasure from the Japanese Emperor. 114 D. Zagier

12 Building Bridges to the East

One of the aftereffects of World War II of which Hirzebruch was most painfully aware was the Cold War division of Europe into East and West, and throughout his life he did what he could to counteract this split. This applied above all to “East Germany” (more properly the German Democratic Republic, or GDR), but also to Russia, Poland and the other countries behind the Iron Curtain. Hirzebruch’s involvement with East Germany was marked by a striking symmetry: in 1962 he was the chairman of the Deutsche Mathematiker-Vereinigung (German Mathematical Society) when it was broken up into an only West German society and a new “Mathematical Society of the GDR,” and it was he again who held the same office in 1990 after the fall of the Wall and who was able to preside over the reunification after 28 years of the two estranged siblings. (See [3].) During the beginning of his first chairmanship, the DMV still had both East and West German members, but the Wall had already been built (in 1961) and there was no place where the members from the GDR, from West Berlin and from West Ger- many could come together for their meetings. Hirzebruch’s elegant solution to this was to simply hold the entire program twice, in East and in West Berlin. After the society had been split, he could do nothing more in his DMV role, but during the entire period until reunification he was very active in many other ways. His main concern was the gradual freezing of contacts, which he tried to prevent by inviting GDR mathematicians to the Arbeitstagungen, by many personal visits and lectures in universities and academies there—the Hirzebruch family remembers him often shopping for presents in Bonn the morning before boarding the train—and by individual invitations to the SFB and later to the MPI. Mostly these invitations could not be realized, but there was some modest success with the Arbeitstagung invitations: among the people who came were Herbert Kurke, Wolfgang Vogel, Eberhard Zeidler, Thomas Friedrich, Jens Franke, Jürgen Leiterer and Werner Müller. Some- times mathematicians who came as visitors remained in the West and were helped by Hirzebruch to find positions. During the years when many West Germans considered the country beyond the Wall as too forbidding and too complicated to visit, he never let himself be discouraged. In the words of Eberhard Zeidler, “er hat den Faden nie abreißen lassen”—he never let the thread be severed. When Hirzebruch became president of the DMV for the second time 28 years later, the situation was wildly different. The wall had just fallen, and during the first meeting over which he presided he could proudly announce the reunion of the two mathematical societies into one. But then following the reunification of Germany there was a new problem: the reintegration of the former East German mathematicians into the new system. This turned out to be a gigantic task, but one for which Hirzebruch was especially suited, since he was one of the few people known and trusted by both East and West. The smaller problem concerned the DMV itself, in particular since the Mathematical Society of the GDR had had many more school teachers and in- dustrial mathematicians among its members than was customary in the West. But the major problem was that a very large number of positions, including assistant positions, had been permanent in the East and suddenly no longer were. In particular, the Karl Weierstrass Institute of the GDR Academy of Science employed nearly 200 Life and Work of Friedrich Hirzebruch 115 mathematicians, many in pure research positions, whereas in West Germany, with a three times larger population, there were practically no positions of this sort apart from the four directors of the Max Planck Institute. For a long time, Hirzebruch trav- eled every week to Berlin where he tried—together with the Max Planck Society, the Fraunhofer Society, and members of the federal and state governments—to find solutions for the GDR scientists and working groups. There were many different answers: some of the mathematicians were well known internationally and easily got professorships or research positions in Germany or abroad, others were near retirement and could obtain temporary positions to bridge the remaining years, and yet others found employment in schools or industry. Special programs were set up, for example 28 working groups in different fields that were financed for five years by the Max Planck Society and attached to a university, in the hope that the groups could later be integrated into the university. But there was not a solution for everyone and some remained jobless. Hirzebruch was never reconciled with this situation and suffered sleepless nights where fate had been unkind. Today these problems are a thing of the past and there is hardly any difference between East and West. With some 20 new Max Planck Instiutes in the new federal states, with vivid scientific exchange and easy access by train or airplane, it seems that the only remaining distances are geographical. Another major concern of Hirzebruch during the Cold War years was to keep contact with mathematicians from the countries behind the Iron Curtain. Every year, he invited leading mathematicians from the Soviet Union and other Eastern Euro- pean countries to the Arbeitstagung. With the exception of the year 1967, when no fewer than five Soviet mathematicians (Anosov, Manin, Postnikov, Shafarevich and Venkov) were able to come, these invitations were nearly always declined, but Hirze- bruch later learned that they had nevertheless helped the invitees by making the So- viet authorities aware of their international visibility. Whenever possible, Hirzebruch accepted invitations to Moscow and Leningrad, mostly by the Russian Academy of Sciences, where he gave lectures, participated in conferences, and above all met Rus- sian colleagues and exchanged news about scientific advancements. He was elected a foreign member of the Academy of Science of the USSR in 1988, received the Lobachevsky Prize in 1990 and the Lomonosov Gold Medal in 1996. He was also actively involved in the exchange with the Euler International Mathematical Institute in St. Petersburg, which was formed in those years as part of the Steklov Institute and developed many relations to Berlin and Germany. Eventually, of course, the situation changed radically, and today it is impossible to walk through the corridors of the MPI and not hear Russian spoken. Especially in the years after the fall of the Iron Curtain, Hirzebruch was also very engaged in supporting mathematics in Poland. In 1990, as president of the European Mathematical Society, he wrote a very strong letter of support for the Banach Center in Warsaw, which was struggling financially and politically. This led to the founding of a Banach Center Council in 1993, with Hirzebruch as its chairman for the first eight years, during which he visited the country yearly and was indefatigable in his support at every level. A very personal account of the story can be found in the article by Stanisław Janeczko in [16]. His contribution was greatly appreciated and was honored in 1998 by an international conference on the occasion of his 70th birthday 116 D. Zagier and in the following year by the award of the Stefan Banach Medal of the Polish Academy of Sciences, of which he had become a member in 1997. A second birthday conference in his honor, which he planned to attend until prevented by an accident, was held in Warsaw just before his death in May 2012.

13 “More and More Number Theory in Topology”

The slogan “More and more number theory in topology” that I have taken as the title of this section is due to Hirzebruch, who proposed it “as a theme (familiar to most topologists) under the general title ‘Prospects of mathematics’ ” in his lovely survey paper [1]. Of course his work had already had several points of contact with number theory, for instance through the appearance of Bernoulli numbers in his work on genera and the signature theorem or, at a deeper level, through his proportionality principle, which was used by him and others to compute dimensions of spaces of automorphic forms. But starting at the end of the 60’s the connections grew much closer. This arose in particular from his work on singularities,1 plumbing construc- tions and exotic spheres, which were treated in a course he gave and published as a joint Springer Lecture Notes with K.H. Mayer in 1968. In these notes he discussed the beautiful singularities + + = ∈ Cn 1 a0 +···+ an = = ∈ Nn 1 Xa (z0,...,zn) z0 zn 0 a (a0,...,an) that had been studied by his student Egbert Brieskorn, who had discovered in par- 9 ticular that the 7-dimensional varieties X2,2,2,3,6k+1 ∩ S give an explicit realization of the 28 exotic structures on S7 found by Kervaire and Milnor as k ranges over all integers modulo 28. In this connection, Brieskorn had shown that the signature of the smooth complex n-dimensional variety Xa(ε) obtained by replacing “= 0” by “= ε” in the deﬁnition of Xa is given by n+1 t0 tn sign Xa(ε) = t ∈ Z 0

1A wonderful exposition of Hirzebruch’s work on singularities was given by Brieskorn in [9]. Life and Work of Friedrich Hirzebruch 117 his supervision, since my Oxford supervisor Michael Atiyah had left for Princeton. Hirzebruch, to whom Atiyah had also written with the same suggestion, invited me to Bonn in May 1970 (a memorable occasion for me, as I met both him and my future wife, then Silke Suter, on the same day), spent an entire afternoon discussing these matters with me and Brieskorn, who was just visiting Bonn from Göttingen, and invited me to come both to the Arbeitstagung in June and to Bonn as his Ph.D. student in the Fall. The next phase of Hirzebruch’s involvement with number theory was partly in tandem with me. We found that the cotangent formula I had sent to him could indeed be obtained by a suitable application of the equivariant signature theorem, and more generally that one could find many further topological situations, involving specific low-dimensional manifolds with explicit group actions, where the signature or other topological invariants could be computed both directly from their definitions and also by applying various index theorems, yielding wildly sophisticated proofs of elementary number-theoretical identities. Hirzebruch presented these results in lectures that he gave in Japan and then in a course in Bonn that we published as a joint book “The Atiyah-Singer Index Theorem and Elementary Number Theory” in the nicely named series “Publish or Perish” of Michael Spivak, who was then a visitor at the SFB. This work connected topology in sometimes surprising ways with Dedekind sums, class numbers, periodic continued fractions, and Markoff numbers, all very venera- ble themes of classical number theory. A nice result that combined the first three of these themes, based on a theorem of Curt Meyer expressing the value at s = 1of a certain L-series in terms of Dedekind sums, relates the class numbers of real and quadratic fields to continued fractions:√ if p>3 is a prime congruent to 3 (mod 4) for Q which the real quadratic field ( p)has class number 1, and if (b1√,...,br ) denotes the minimal period of the continued fraction√ with negative signs of p (see next sec- − Q − 1 r − tion), then the class number h( p) of ( p)equals 3 i=1(bi 3). A planned joint paper presenting this and related results unfortunately never got written, though the class number formula was stated in papers of each of us separately.

14 Hilbert Modular Surfaces and the Resolution of Their Cusp Singularities

Among the various specific low-dimensional manifolds that Hirzebruch and I had studied for our applications of index theorems to number theory were torus bundles over a circle. These led directly to the far more exciting second phase of Hirzebruch’s life as a number theorist, his study of the geometry of Hilbert (or Hilbert-Blumenthal) modular surfaces, to which I now turn. This investigation had three main parts: the resolution of the singularities of Hilbert modular surfaces at their cusps, the computation of the numerical invariants of these surfaces and its application to determining their type (rational, K3, elliptic, or general type, or equivalently Kodaira dimension −∞, 0, 1 or 2) in the classification of algebraic surfaces, and the study of the intersection behavior of the modular curves lying on these modular surfaces. I will describe all three briefly. Hilbert modular surfaces are a generalization of the modular curve H/SL(2, Z) (H = complex upper half-plane) from the classical theory of modular forms. To a 118 D. Zagier real quadratic field K one associates (at least in the simplest case; there are many 2 variations) the quotient XK = H /SL(2, O).Here O = OK is the ring of integers of K and SL(2, O) is the group of 2 × 2 matrices ab with coefficients in O and cd 2 az1+b a z2+b determinant 1, acting on H by (z1,z2) → ( , ), where x → x denotes the cz1+d c z2+d Galois conjugation of K over Q. This surface has a natural compactification by the addition of hK (= class number of K) points called the cusps. A neighborhood of the ={ ∈ 2 | } “cusp at infinity” is the quotient of the set WC (z1,z2) H (z1) (z2)>C , ab where C is a sufficiently large constant, by the group of triangular matrices 01/a with a ∈ O∗ and b ∈ O. For other cusps or other Hilbert modular surfaces (e.g. those obtained from subgroups of SL(2, O) of finite index) one has more generally the quo- = εμ | ∈ ∈ tient WC/G(M,V), where G(M, V ) 01 μ M,ε V for some sublattice M =∼ Z2 ⊂ K and some infinite cyclic group of totally positive units V acting on M. The problem is then to find a smooth open surface YC(M, V ) containing a divisor consisting of smooth curves with normal crossings whose complement is isomorphic to WC/G(M,V). The solution of this problem was in some sense an outgrowth of Hirzebruch’s doctoral thesis in which, as we saw, he had studied the resolution of 2-dimensional singularities and had shown that they are sometimes described number-theoretically = n q in terms of continued fractions, as in the example for the singularity w z1 /z2 that we described earlier. (The necessity of using continued fractions with minus signs, rather than the traditional ones with plus signs, was one of his insights.) The solution of the problem for the cusp singularities turned out to be related. To the cusp labeled by (G, V ) as above one associates (by taking w ∈ K equal to the ratio of the two elements of a suitable Z-basis for M)aperiodic continued fraction expansion whose partial quotients bi are uniquely determined up to cyclic permutation by the pair (M, V ). Hirzebruch then showed that the resolution of the singularity is given by the corresponding cycle of rational curves, with self-intersections −bi just as for the finite continued fractions that had occurred in his thesis:

He thus made spectacular amends for the wrong opinion stated there by showing not only that cycles can occur in the resolution of 2-dimensional singularities, but that a major problem in number theory could be solved using precisely these forbidden cycles! We will describe here the way this cusp resolution works, because it is both a beautiful piece of mathematics and a beautiful example of Hirzebruch’s seamless fusion of ideas from topology, algebra, analysis, geometry and number theory. We embed M as a lattice in R2 via x → (x, x ). Then the boundary of the convex hull of ∩R2 M >0 consists of the line segments joining the images of countably many elements Ak ∈ M (k ∈ Z), of which any two successive ones form a Z-basis of M and any three Life and Work of Friedrich Hirzebruch 119 successive ones are related by bkAk = Ak−1 + Ak+1 for some integer bk ≥ 2, with at least one bk ≥ 3.

The action of a generator ε of V by multiplication on M sends Ak to Ak+r for some r>0 and all k,sothebk are periodic (bk+r = bk; r is in fact the minimal period of the bk multiplied by the index of V in the group of all totally positive units preserving M). Then the quotients wk = Ak/Ak+1 satisfy wk = bk − 1/wk+1 and hence all are given by periodic continued fractions like that of w = w1 above, with periods related by cyclic permutations. The local ring R of the singularity consists of G(M, V )-invariant holomorphic functions F(z1,z2) on WC for sufficiently large C. Every such function, and in particularly every Hilbert modular form, has a Fourier expansion of the form 2πi(νz1+ν z2) F(z1,z2) = c(0) + c(ν)e , ν∈M∨,ν0 ∨ ∨ 1 where M ={ν ∈ K | tr Q(νM) ⊆ Z} is the dual of M (e.g. M = √ O for K/ D = O = M , where D is the discriminant of K) and where c(εν) c(ν) for all ν.For each k ∈ Z we can write F(z ,z ) as a power series c(ν)utr(Akν)vtr(Ak−1ν) in the 1 2 k k ∈ C∗2 C2 z1 = Ak Ak−1 log uk coordinates (uk,vk) on /M defined by 2πi z .This 2 Ak Ak−1 log vk identifies R with a subring of the product of r copies of the ring of somewhere con- vergent power series in two variables: Z Z − R =∼ ∈ C{ } /r = bk 1 (Fk)k(mod r) u, v Fk(u, v) Fk+1 u vk,uk for all k . Since each point on the surface is near the origin in at least one coordinate system ∈ R2 (uk and vk are bounded by 1 in absolute value if the point ( (z1), (z2)) >0 belongs to the closed cone spanned by Ak−1 and Ak), this description of the local ring corresponds geometrically to a resolution of the singularity by a cycle of rational curves Sk as above, where Sk is given by the equation vk = 0inthekth coordinate system (uk,vk) and by uk+1 = 0inthe(k + 1)st coordinate system, and where the intersection point of the curves Sk−1 and Sk corresponds to the origin (uk,vk) = 0 in the kth coordinate system. The self-intersection of Sk is −bk because the divisor of the meromorphic function represented by uk in the kth coordinate system contains Sk−1, Sk and Sk+1 with multiplicities bk, 1 and 0, respectively, and has intersection number 0 with Sk. 120 D. Zagier

Hirzebruch’s resolution of the cusp singularities was one of the first non-trivial examples of what are now called toroidal embeddings and toroidal compactifications, and was one of the motivations for the development of this theory by Mumford, Sa- take, and others. Another offshoot of the cusp resolution was a formula that I found for the decomposition of the Dedekind zeta function of K and that was generalized by Shintani to arbitrary totally real number fields. The second problem that Hirzebruch tackled was the determination of how the Hilbert modular surfaces fit into the Enriques-Kodaira rough classification of surfaces. A key step here was the calculation of the numerical invariants of the surface, which in turn had three ingredients: the cusp resolution, a formula for the Euler characteristic in terms of ζK (2) (the value at s = 2 of the Dedekind zeta function of the quadratic field) using a Gauss-Bonnet theorem for non-compact manifolds due to Günter Harder, whom Hirzebruch had brought to Bonn as a colleague, and the signature theorem. All Hilbert modular surfaces belong to one of four classes in the classification (rational, K3, non-K3 elliptic fibrations, or general type), and Hirzebruch succeeded in classifying all of them, doing the cases when K has prime discriminant with Van de Ven and the case of general discriminant with me. The result for prime discriminants is particularly attractive: XK is rational for the first three values 5, 13 and 17, is a K3 surface for the next three values 29, 37 and 41, is a non-K3 elliptic fibration for the next three values 53, 61 and 73, and is of general type for all larger prime discriminants. For arbitrary discriminants one finds that there are precisely 10, 9 and 13 surfaces having the three special types. Hirzebruch’s last big contribution to the theory of Hilbert modular surfaces concerned the study of certain curves on these surfaces and their intersection behavior, and involved another beautiful interplay between topology and number theory. This investigation was carried out jointly with me, and for me remains perhaps the most exciting collaboration I have ever been involved with. For each positive integer N one defines a curve TN on XK as the union of all points satisfying equations of the form ∨ az1z2 + νz1 + ν z2 + b = 0 with a,b ∈ Z, ν ∈ O and ab − νν = N/D. Hirzebruch found a geometric way to describe their intersection points, and together we were able to use this to get an exact formula for the finite part of the intersection numbers, based on the theory of complex multiplication and involving some fairly serious algebraic number theory. Then using his resolution of the cusps we could also compute the intersection numbers of the corresponding curves TN on the resolved and com- pactified surface. The high points were the realization that these numbers were the Fourier coefficients of modular forms (this involved recognizing the modular nature of the generating function of class numbers of imaginary quadratic fields, which I did and which later turned out to be the first example of what are now called mock modular forms) and our main theorem identifying the relevant part of the cohomology of the Hilbert modular surface with a certain space of modular forms of weight 2. This theorem was a prototype for many later generalizations in which the generating series of some geometric invariants turn out to be modular forms. An amusing story in this connection is that at one point in this study a letter from Serre to Hirzebruch, asking whether he could explain the numerical coincidence between a certain Betti number and the dimension of this space of modular forms, crossed in the post a letter from Hirzebruch to Serre in which he described our work and answered this very question! Life and Work of Friedrich Hirzebruch 121

Finally, I should mention again Hirzebruch’s conjecture on the signature defect and its relation to special values of L-functions for higher number ﬁelds, which was later solved by Atiyah–Donnelly–Singer and independently by Werner Müller. Hirzebruch wrote up the whole theory in a long article in L’Enseignement Mathé- matique that also appeared as a monograph, and then later in a second book with Gerard van der Geer based on a joint course that they had given in Montréal. This developed into a close friendship with van der Geer, with whom he later organized many meetings in the mountain village of Alpbach in the Austrian Tirol, where the two of them and a group of undergraduates would spend a fortnight alternating between intensive mathematical and intensive hiking activities.

15 The Founding of the Max Planck Institute for Mathematics

We have already told of Hirzebruch’s first abortive attempts to found a Max Planck Institute of Mathematics, and of the successful founding of the Sonderforschungs- bereich Theoretische Mathematik instead. But the lifetime of a Sonderforschungs- bereich was normally 6 or 9 years, and could certainly not exceed 15. So already in 1977/78 Hirzebruch discussed with the third Max Planck president, Reimar Lüst, the prospects of a new attempt at establishing an MPIM. This time the auspices were more favourable: not only was there a very successful SFB to build upon and which seemed to meet a real demand, but Hirzebruch had also kept his full teaching load at the university, and moreover, the costs of the institute would be modest since there were no guest researchers to transfer into permanent positions. In a letter to Lüst that was only three and a half pages long, Hirzebruch outlined the structure and func- tioning of the future institute, as well as a program for the period of transition of the Sonderforschungsbereich into the MPIM. His letter formed the basis both for discussions inside the MPS and later, in 1979, for a meeting organized by the Alexander von Humboldt Foundation, to which Saunders McLane and other prominent mathematical scientists from different fields of mathematics were invited and where Reimar Lüst was present. The discussion there centered on the situation of mathematics in Germany, and soon Hirzebruch realized that the meeting was meant as a launching pad for the project of founding an MPI in mathematics. Indeed, in the following year, the senate of the MPS decided to found the MPIM. There followed a transition period of three years, during which the financing of the visitors’s program was covered by the SFB and the MPIM in the ratios 75:25, 50:50 and 25:75, respectively. By 1984 the MPIM had taken up its full activities and the institute had moved from its old quarters in Beringstraße to a rented office building in Beuel on the other side of the Rhine (Fig. 4). The full story is recounted by Norbert Schappacher in [2]. According to plan, there were very few permanent positions—in the beginning only Hirzebruch full time, myself half-time (the other half being spent in Maryland), and Harder as an External Scientific Member. Following the Princeton model, most of the guests were young researchers, mainly post-docs, from all over the world, with the Japanese and the Russian groups standing out in size and influence, as we have already described. As the number of visitors grew, so did the scientific and non- scientific personnel. I became a full member in 1984 and Harder in 1991. We were 122 D. Zagier

Fig. 4 The president of the Max Planck Society, Reimar Lüst, with F. Hirzebruch at the inauguration of the Max Planck Institute for Mathematics on June 24, 1982

joined by Yuri Manin in 1993 and Gerd Faltings in 1994, so that by the time that Hirzebruch retired in 1995 the institute had a Director and four Scientific Members. Since none of the four of us felt capable of assuming his role, we suggested to the Max Planck Society that we should all share the title and duties of director, with the role of managing director rotating among us on a two-year basis, and this has been the model ever since, with Werner Ballmann and Peter Teichner later joining the institute as the successors of Harder and Manin, respectively. In the course of time, there were a few further long-term positions, most notably the special professorship that was created for Matilde Marcolli (gauge theory, mathematical physics, non-commutative geometry), who joined the MPIM from 2000 and 2008, won the Leibniz prize for the Institute, and was indefatigable in arranging research activities and in taking care of young students from the IMPRS graduate school program that started in 2003. The Institute now also has three External Scientific Members, Werner Nahm since 2006 and Sergei Gukov and Stefan Müller since 2010. In 1999, the MPIM moved from Beuel to its current premises in the former “Post- carré” in the center of Bonn, creating much more space for offices, seminar rooms and library and making it easier to commute to the University. The atmosphere is extremely lively and international: in [8], Hirzebruch says that he counted on a random day and found 24 nationalities among 79 researchers, with the most frequently represented being Russia, Germany, the USA, France, Japan, Israel, Great Britain, Life and Work of Friedrich Hirzebruch 123 and Iran, in that order. As well as the Arbeitstagung and many long-term activities in various fields, there is an ongoing flow of seminars and talks, many of which are organized together with the university, as well as a cooperation with the Hausdorff Institute for Mathematics, which was founded in 2006 as part of the Hausdorff Cen- ter for Mathematics, to which the MPIM also belongs. A further very successful project, also closely coordinated with the university, is the already-mentioned graduate school, which is perhaps the most visible proof that the MPIM does not work in an ivory tower.

16 Hirzebruch and Israel

A seldom mentioned, but always present theme in Hirzebruch’s life was his acute awareness of the German crimes under the Nazi regime. One of the rare occasions when these feelings were made explicit was his acceptance speech of the Wolf Prize in the Israeli Knesset in 1988. I was present on that occasion and will never forget the emotions that I felt myself and saw in the audience on hearing the final words of his speech: ...Asa professorattheUniversityofBonn, I am one of the successors of the famous mathematicians Felix Hausdorff and Otto Toeplitz. Hausdorff commit- ted suicide in 1942, together with his wife, when deportation to a concentration camp was imminent; Toeplitz emigrated to Israel in 1939 and died there the following year. The memory of these mathematicians is with me always on this trip. Hirzebruch had an intense relationship with Israel. He first visited the country in 1981, and starting in the late 1980’s became actively involved in the organization of mathematical research there. He was asked and agreed to serve in the Advisory Board of the Gelbart Institute of Bar-Ilan University in Ramat-Gan and in the Beirat (advisory body) of the Edmud Landau Research Institute for Mathematical Analysis of the Hebrew University in Jerusalem. He also served in the Evaluation Committee of the mathematics department in the Technion in Haifa and supported the establishment of the Minkowski Center in Tel-Aviv University. Above all, Hirzebruch had a fundamental role in the Emmy Noether Institute of Bar-Ilan University. In the 1990 meeting of the Advisory Board of the Gelbart Insti- tute (an international visitor program based on a private donation) in Bar-Ilan Uni- versity, he told the others about the Minerva Foundation, a daughter organization of the Max Planck Society that was founded to foster high-level research and had some 40 centers in Israel in all sciences, each co-funded by the Israeli Institute. At his suggestion and following his advice, the university applied for a joint German-Israeli Minerva center in Mathematics, to be named after the German-Jewish mathematician Emmy Noether, who had fled from Germany to the US in the early 1930’s. The application was approved in 1991 and an inauguration ceremony of the Emmy Noether Institute took place in the house of the Israeli Ambassador in Bonn in July 1992. A bi- national Beirat was appointed by the Bar-Ilan University and the German Ministry of Education and Research, with Hirzebruch being appointed as its chairman. He served 124 D. Zagier in this role for twelve years with great success, helping in particular to extend the net- work of the institute, to sign exchange agreements with other institutes (including the Banach Institute in Warsaw), to enlarge the visitor program, and to raise the visibility of the center. The University of Bar-Ilan paid tribute to Hirzebruch’s contribution by granting him an honorary doctorate in 2000, and the Emmy Noether Institute organized two major international conferences in algebraic geometry in his honor, “Hirz 65” in 1993 and “Hirz 80” in 2008, on the occasion of his corresponding birthdays. The latter of these was probably the last big conference he attended: he went to all the talks, discussed mathematics and old times with his many friends there, and participated in all the tours (unfortunately falling and breaking a leg during the final one, to the Western Wall caves, but then apologizing profusely for disturbing the agenda!). More about Hirzebruch’s relations to Israel can be found in the contribution to [16]by Mina Teicher, director of the Emmy Noether Institute, organizer of the two Bar-Ilan conferences, and Hirzebruch’s close friend.

17 “More and More Algebraic Geometry in Combinatorics”

This time the slogan is mine, not Hirzebruch’s, but is still very appropriate, since he wrote a number of papers in the 80’s and 90’s on different links between these two subjects. Of the links he found, perhaps the most beautiful and unexpected was his application of the theory of algebraic surfaces, and in particular of the Miyaoka-Yau inequality, to the classical problem of arrangements of lines in a plane. This problem, which goes back at least to a paper of Sylvester of 1893, but in fact to a problem about planting trees in an orchard that was posed in the form of a poem in the early 19th century, is very easy to state: one asks what the constraints are for the numbers of meeting points of various valencies in a ﬁnite conﬁguration of lines in the plane (“arrangement”). Of course one is interested in cases where the lines meet in many points, of multiplicity as high as possible: Life and Work of Friedrich Hirzebruch 125

The problem is subtle, because there are arrangements that can be realized combinatorially (e.g., in the projective plane over a finite field) but not over C and others (like the 12 lines through the 9 inflection points on a plane cubic) that can be realized over C but not over R. Hirzebruch discovered a beautiful way to apply the 2 ≤ Miyaoka-Yau inequality c1 3c2 for compact complex surfaces to certain ramified coverings of P2(C) associated to a complex line arrangement to obtain a result that is stronger than anything that has been obtained by elementary methods. His inequality reads

d− 3 2 t + t ≥ d + (2i − 9)ti, 2 4 3 i=5 where d is the number of lines in the arrangement and ti (assumed to vanish for i = d − 1ori = d) is the number of points through which precisely i lines pass. This inequality was used, among other applications, to prove the “bounded negativity” 2 2 conjecture (d − c)/n > −4 (where n = ti is the number of points and c = i ti the “self-crossing count”) for line arrangements. In the converse direction, Hirzebruch and his students and coworkers used special line arrangements to construct interesting algebraic surfaces, and in particular surfaces where the Miyaoka-Yau bound is attained. These examples were presented in detail in a book with Gottfried Barthel and Thomas Höfer, and form an important contribution to the “geography problem,” still not completely solved, which asks to 2 describe all possible Chern numbers (c1,c2) of algebraic surfaces. The extreme cases 2 = with c1 3c2 are particularly interesting, since the Hirzebruch proportionality principle implies that this equality holds for quotients of the complex 2-ball by discrete group actions, and the Miyaoka-Yau theorem states that it holds in no other cases. Hirzebruch was extremely interested in such “ball quotients,” which are analogous in many ways to Hilbert modular surfaces, and in particular wrote an 18-page long review, together with Paula Beazley-Cohen, of a book by Deligne and Mostow, in which his examples are linked with those discussed in the book and with the theory of hypergeometric functions of several variables. He continued to occupy himself with this subject for several years and for a long time planned a book on it with Beazley-Cohen, though in the end this joint project was abandoned and work on the book was continued by her alone. But there were also many smaller projects linking combinatorics and algebraic geometry: a paper relating Euler polynomials and “zigzag permutations” to the holomorphic Euler characteristics of certain algebraic varieties, lectures about the icosa- hedron√ (which had also played a role in his investigation of a Hilbert modular surface for Q( 5)) and “Fullerenes,” and his 2009 Oberwolfach lecture in which Catalan numbers and other numbers of combinatorial interest are related to the Chern numbers of Grassmannians and other Schubert varieties. Brieskorn, writing in [9] about Hirzebruch’s later work in algebraic geometry, speaks of “a wealth of beautiful geometry with relations to classical conﬁgurations of the 19-th century, but also to modern theoretical physics.” 126 D. Zagier

Fig. 5 F. Hirzebruch giving a lecture on elliptic genera at the Arbeitstagung 1987

18 Elliptic Genera

In the late 1980’s the theory of elliptic genera was developed by Ochanine, Landweber-Stong, and Witten, and of course Hirzebruch was immediately interested: elliptic genera are the genera, in the sense that he had invented and that we described earlier in this article, having certain classical elliptic functions as their deﬁning power series, so that they attach modular forms to manifolds, thus combining in a deep way two subjects that were very dear to his heart. Moreover, the special values of these modular forms at the two cusps of the modular group in question are precisely the signature and the A-genus that he had studied alone and in his joint work with Atiyah. Witten and others had given motivations coming from physics for this theory using the free loop space of a manifold, and this had led Witten to the formulation of a rigidity conjecture for the elliptic genus, later proved by Taubes and by Bott-Taubes, that directly generalized the earlier rigidity theorem of Atiyah and Hirzebruch for spin manifolds with S1-action. Hirzebruch gave a course on the subject that appeared as a joint book with the students Thomas Berger and Rainer Jung, and also wrote several papers on the subject, including a long joint one “Elliptic genera, involutions, and homogeneous spin manifolds” with Peter Slodowy. Elliptic genera, and some of the results found by Hirzebruch in connection with the course (in particular, a generalization to higher level), have applications in both pure mathematics and mathematical physics, in particular string theory. When asked at the end of his interview with Matthias Kreck [14] whether there were new ﬁelds of mathematics that he would like to learn, Hirzebruch answered: “I would not mind to be an expert in string theory”!

19 The Years after Retirement

Hirzebruch retired from the university in 1993 and as director of the MPIM in 1995, each occasion being celebrated by a party and lectures in his honor, in accordance with his dictum that one should never pass up any opportunity to celebrate. There was also a big party in 1992 to which all 52 of his doctoral students were invited and to which 33 came, some from overseas. Life and Work of Friedrich Hirzebruch 127

During the years following his retirement Hirzebruch remained active both mathematically and in his various national and international roles, and continued to come to the MPI almost every day, when he was not traveling, nearly until his death 19 years later. Much of the mathematics that occupied him during these years was a continuation of earlier directions that have already been described, in particular on relations between combinatorics and algebraic geometry and on the theory of elliptic genera. Here I mention only one other topic, this time from pure number theory (though originally motivated like everything else from topological considerations), which I particularly like because it was joint with me! This was an investigation of “twisted Dedekind sums,” which are analogues of the classical Dedekind sums that we had studied in our book, but now involving a quadratic character. Here in his own handwriting is a formula from this work that he wrote down as a present to his friend Taida Hambleton in 1999 on the occasion of a dinner at his beloved restaurant “Maternus” in Bad Godesberg:

Among other things, we found that the twisted Dedekind sum

1 k πk πkd f(p,d)= √ cot cot p ≡ 1 (mod 4) prime,p d 4 p √ p p p 0

h(−3p) f(p,1) = 2f(p,2) = 3f(p,3) + = 4f(p,4) + h(−4p) = 6ζK (−1), 2 √ where ζK (−1) is the value at s = 1 of the Dedekind zeta function of K = Q( p) 1 Z − − (and belongs to 3 if p>5), and h( 3p) and h( 4p) are class numbers. These formulas are reproduced in Hirzebruch’s article [8]. A planned joint paper never got written, but the results will be included in a greatly expanded version of our joint book that is currently in preparation, with Paul Gunnells as a third co-author. We already discussed some of the international tasks that Hirzebruch continued to perform also after his retirement, such as the chairmanship of the Beirat of the Emmy Noether Institute from 1992 till 2004 and the chairmanship of the Banach Center Council from 1993 till 2001, and of course he also continued to travel and lecture widely. In 1990 he became the first president of the newly formed Euro- 128 D. Zagier pean Mathematical Society, and served in that role until the end of 1994. Perhaps the high point came in 1998, when he was asked to serve as honorary president of the International Mathematical Congress held in Berlin, the first one to take place in Germany in 94 years, and gave a moving opening address in which he reminded his audience once again of the tragedies of the Nazi era and announced the special activity organized by the German Mathematical Society to honor the memory of the victims. Many honors had of course come to him even before retirement, but in these later years they multiplied. We have already mentioned the conferences that were held in his honor in Israel and in Poland. His first six honorary doctorates arrived at roughly the same period and at roughly the same frequency as his six grandchildren, so that he would often say jokingly, depending on which side was ahead, that it was time for a new grandchild or a new doctorate—but it was an unequal battle and in the end he received a total of fifteen honorary doctorates, from Warwick, Göttingen, Ox- ford, Wuppertal, Notre Dame, Dublin, Athens, Potsdam, Konstanz, Berlin, Bar-Ilan, Oslo, Chicago, Bucharest and Augsburg. Then there were the prizes. We have already spoken of the Wolf Prize from Israel, the Lobachevsky prize and Lomonosov Gold Medal from Russia, the Stefan Banach Medal from Poland, and the Seki prize and the Order of the Holy Treasure from Japan, but there were many more, including the Medal of the City of Paris and the Albert Einstein Medal of Switzerland and many from Germany: the Knight Commander’s Cross of the Federal Republic of Germany, the Golden Cothenius Medal of the Leopoldina Academy, the Krupp Science Prize, the Helmholtz Medal of the Berlin-Brandenburg Academy of Science, and the Georg Cantor Medal of the Deutsche Mathematiker-Vereinigung. He was also a member or foreign member of more than twenty German and international academies, of which he attached particular importance to the order “Pour le Mérite” because of the extraordinary fellow members that he could get to know at their meetings, almost all of which he attended. In the last years of his life, Hirzebruch had a series of major and minor health problems, including two serious falls, one at the end of the “HIRZ80” conference in Tel Aviv that we have already mentioned and one only a few weeks before his death. He seldom spoke of these problems and continued, not only to come to his office in the Max Planck Institute nearly every day, but to maintain his interest in both mathematics and mathematicians, making it a point to get to know almost every visitor personally. Five weeks before his death in May 2012, he gave a beautiful two-hour lecture on “The shape of planar algebraic curves defined over the reals” at the MPI that the interested reader can hear on the MPI’s on-line “Hirzebruch Collec- tion.” His death was reported on by obituaries in numerous newspapers in Germany and abroad. Many people, some coming from far away, came to his funeral to show their affection and pay their respects to a man who had affected all of their lives (Fig. 6). Half a year before his death, Hirzebruch gave an in-depth interview [14]forthe Simons foundation, with his former student Matthias Kreck as the interviewer, that gives a vivid impression of his life and personality. The accompanying article by Joel Segel [15] has the title “Friedrich Hirzebruch, Giant of German Mathematics.” There is no better description imaginable. Life and Work of Friedrich Hirzebruch 129

Fig. 6 F. Hirzebruch’s last lecture on “The shape of planar algebraic curves deﬁned over the reals” in May 2012

20 Writings by and about Hirzebruch

Hirzebruch’s skills as an expositor of mathematics were just as evident in his writing as in his lectures. It was always beautifully clear, almost entirely free of errors, and written in an informal and very human style that was nevertheless at the same time also extremely precise. Here, for instance are some quite typical sentences from the write-up of his 2009 Oberwolfach lecture about Chern classes, in which, after explaining how to calculate the number of lines on a cubic hypersurface in P3, a quintic hypersurface in P4, or a degree 7 hypersurface in P5, he adds: The number of lines on a hypersurface occurred as a Chern number. We had to neglect, for example, the long history of the discovery of the 27 lines on a cubic surface. On my desk there is the classical model of a cubic surface of Clebsch and Klein defined over the reals. Here also the 27 lines are defined over the reals. Thus I can see them any time I wish. In this special case the lines have 10 triple points. Altogether Hirzebruch published well over a hundred mathematical articles, mostly research but also many expositions of his own work or that of others, including several exposés in the Séminaire Bourbaki. We will not give the full list here, since it is too long and can be found on the Hirzebruch Collection page on the MPIM website (https://hirzebruch.mpim-bonn.mpg.de). We do, however, give a complete list of his many books (which include not only the famous Topological Methods but also a number of research monographs or textbooks, often based on courses he gave and sometimes co-authored with people who had attended them), followed by a list of the mathematical articles that were discussed in the main text, together with their numbers in his published Collected Papers. In the following and final paragraph “References” we then give a short list, in chronological order, of various more historically-oriented articles written by or about him.

Books by Friedrich Hirzebruch New Topological Methods in Algebraic Geometry [German, 1956 and 1962; En- glish 1966; Japanese, 1970; Russian, 1973; Chinese 2004. Springer “Classics in Mathematics” 1995] Sheaf- and Cohomology Theory (with G. Scheja) [German, 1957] 130 D. Zagier

Introduction to the Theory of Vector Bundles and K-Theory (notes by M. Haze- winkel and D. Erle) [1965] O(n)-Manifolds and Exotic Spheres (with K.H. Mayer) [German, 1968] Introduction to Functional Analysis (with W. Scharlau) [German, 1971 and 1996] Differentiable Manifolds and Quadratic Forms (with W.D. Neumann und S. Koh) [1971] Hilbert Modular Surfaces [1973] The Atiyah-Singer Theorem and Elementary Number Theory (with D. Zagier) [1974] Lectures on Hilbert Modular Surfaces (with G. van der Geer) [1981] Numbers (with H.-D. Ebbinghaus et al.) [German, 1983 and 1992; English, 1996] Line Conﬁgurations and Algebraic Surfaces (with G. Barthel and Th. Höfer) [Ger- man, 1987] Manifolds and Modular Forms (with Th. Berger and R. Jung) [1992 and 1994]

Articles by F. Hirzebruch that are discussed individually in the text: Note: Numbers like “I.10” or “II.43” refer to papers from Volume I (papers 1–33) or Volume II (papers 34–75) of the Gesammelte Abhandlungen. Numbers starting with “III” refer to the informal Volume III (papers 76–93 and a.–g.) that was produced at the MPI in 2001, but since this volume is not published, complete references have been given for these papers. It is planned to prepare and publish a more complete Volume III. I.1. Über eine Klasse von einfach zusammenhängenden komplexen Mannigfaltig- keiten (1951). I.2. Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichern (1953). I.10. Some problems on differentiable and complex manifolds (1954). I.M2. Neue topologische Methoden in der algebraischen Geometrie (1956). I.16. Automorphe Formen und der Satz von Riemann-Roch (1958). I.19/22/26. (with A. Borel) Characteristic classes and homogeneous spaces. I, II, III (1958/59/60). I.27/32. (with M.F. Atiyah) Vector bundles and homogeneous spaces (1961); (with M.F. Atiyah) Charakteristische Klassen und Anwendungen (1961). I.21/24. A Riemann-Roch theorem for differentiable manifolds [Bourbaki] (1959); (with M.F. Atiyah) Riemann-Roch theorems for differentiable manifolds (1959). I.25. (with M.F. Atiyah) Quelques théorèmes de non-plongement pour les variétés différentiables (1959). I.27. (with M.F. Atiyah) Vector bundles and homogeneous spaces (1961). I.28. (with M.F. Atiyah) Bott periodicity and the parallelizability of the spheres (1961). I.32. (with M.F. Atiyah) Charakteristische Klassen und Anwendungen (1962). II.37. Elliptische Differentialoperatoren auf Mannigfaltigkeiten (1966). II.38. Über Singularitäten komplexer Flächen (1966). II.39. Singularities and exotic spheres [Bourbaki] (1967). Life and Work of Friedrich Hirzebruch 131

II.43. (with M.F. Atiyah) Spin-manifolds and group actions (1970). II.45. Free involutions on manifolds and some elementary number theory (1971). II.46. Pontrjagin classes of rational homology manifolds and the signature of some affine hypersurfaces (1971); Pontrjagin classes of rational homology manifold [Notes by S. Morita] (1972). II.48./51. The Hilbert modular group, resolution of the singularities at the cusps and related problems [Bourbaki] (1971); Hilbert modular surfaces (1973). II.53. (A. Van de Ven) Hilbert modular surfaces and the classification of algebraic surfaces (1974). II.60. (with D. Zagier) Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus (1976). II.61. (with D. Zagier) Classification of Hilbert modular surfaces (1977). II.69. Arrangements of lines and algebraic surfaces (1983). II.73. Algebraic surfaces with extreme Chern numbers (report on the thesis of Th. Höfer) (1985). III.79. (with P. Slodowy) Elliptic genera, involutions, and homogeneous spin manifolds. Geometriae Dedicata 35 (1990) 309–343. III.82. Mannigfaltigkeiten und Modulformen. Jber. d. DtMath.-Verein.˙ (1992), 20– 38. III.84. (with P. Beazley-Cohen) Book review: Commensurabilities among lattices in PU(1,n), by Pierre Deligne and G. Daniel Mostow. Bull. AMS 32 (1995) 88–105.

References

1. Hirzebruch, F.: The signature theorem: reminiscences and recreation. In: Prospects in Mathematics. Ann. Math. Stud., vol. 70, pp. 3–31 (1971) 2. Schappacher, N.: Max-Planck-Institut für Mathematik, Historical notes on the new research institute at Bonn. Math. Intell. 7, 41–52 (1985) 3. Hirzebruch, F.: Centennial of the German Mathematical Society. In: Hilton, P., Hirzebruch, F., Rem- mert, R. (eds.) Miscellanea Mathematica, pp. 177–194. Springer, Heidelberg (1991) 4. Hirzebruch, F.: German-Russian cooperation in mathematics. Mitt. Dtsch. Math.-Ver. 4, 54–58 (1997) 5. Hirzebruch, F.: Learning complex analysis in Münster–Paris, Zürich and Princeton from 1945 to 1953. Gaz. Math. 74, 27–39 (1997) 6. Hirzebruch, F.: Kunihiko Kodaira: mathematician, friend and teacher. Not. Am. Math. Soc. 45, 1456– 1462 (1998) 7. Hulek, K.: Friedrich Hirzebruch and mathematics in post-war Germany. In: Yau, S.T. (ed.) The Founders of Index Theory, pp. 203–221. International Press, Somerville (2004) 8. Hirzebruch, F.: Gründungsgeschichte des Max-Planck-Instituts für Mathematik. Mitt. Dtsch. Math.- Ver. 14, 73–79 (2006) 9. Brieskorn, E.: Singularities in the work of Friedrich Hirzebruch. Surv. Differ. Geom. 7, 17–60 (2007). (Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer) 10. Hirzebruch, F.: The Atiyah-Bott-Singer ﬁxed point theorem and number theory. Surv. Differ. Geom. 7, 313–326 (2007). (Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer) 11. Hirzebruch, F.: The ﬁrst Arbeitstagungen with special emphasis on 1957, 1958 and 1962. MPIM Preprint series No. 2007-75-a. http://www.mpim-bonn.mpg.de/node/263 12. Hirzebruch, F.: Bericht über meine Zeit in der Schweiz in den Jahren 1948–1950. Schweizerische Mathematische Gesellschaft, 303–315 (2010) 13. Hirzebruch, F.: Why do I like Chern, and why do I like Chern classes? Not. Am. Math. Soc. 58, 1231–1234 (2011) 14. Kreck, M.: Video interview with Friedrich Hirzebruch. Simons Foundation website (2011) 132 D. Zagier

15. Segel, J.: Friedrich Hirzebruch: Giant of German Mathematics. Science Lives Project. Simons Foun- dation website (2011) 16. Atiyah, M.F., Zagier, D. (coordinating editors): Friedrich Hirzebruch (1927–2012). Not. Am. Math. Soc. 61, 2–23 (2014) 17. Atiyah, M.F.: Friedrich Ernst Peter Hirzebruch, 17 October 1927–27 May 2012, Biogr. Mem. Fellows R. Soc. 60, 229–247 (2014)

Don Zagier is an American mathematician who came to Germany in 1970 to complete his Ph.D. under the supervision of Friedrich Hirze- bruch and ended up spending essentially his entire professional life there, though often with parallel positions in other countries (USA, Hol- land, France, Italy). He began working in topology but then moved to number theory, especially the theory of modular forms, and connections to other ﬁelds of mathematics and mathematical physics. His work with Hirzebruch, ﬁrst on relations between index theory and elementary number theory and then on Hilbert modular surfaces, was decisive throughout his mathematical life. Jahresber Dtsch Math-Ver (2015) 117:133–160 DOI 10.1365/s13291-015-0113-2 SURVEY ARTICLE

Combined First and Second Order Variational Approaches for Image Processing

Gabriele Steidl1

Published online: 10 March 2015 © Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2015

Abstract Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like image restoration, registration, segmentation, super-resolution, and estimation of flow fields. We review recent progress in mathematical image processing by combining first and second order derivatives in the regularization term of variational models. We demonstrate the power of the proposed methods by two rather different applications. The approaches make use of two different splitting methods of the functional to obtain iterative numerical schemes which require in each step only the computation of simple proximal mappings.

Keywords Mathematical image analysis · Variational models · Primal-dual algorithms · Higher order models · Optical ﬂow · Strain tensor · Cyclic proximal point algorithm · Cyclic data

Mathematics Subject Classiﬁcation 49J40 · 49M29 · 49M37 · 49N45 · 52A41 · 65K10

1 Introduction

Mathematical image processing is an innovative, rapidly developing branch of applied mathematics which involves various mathematical ﬁelds as applied harmonic analysis, inverse problems, numerical analysis, partial differential equations, mathematical morphology, probabilistic and statistical methodologies, and variational methods. In this paper we focus on the later one. More precisely, we will consider

B G. Steidl [email protected]

1 University of Kaiserslautern, Dept. of Mathematics, Paul-Ehrlich-Str. 31, 67653 Kaiserslautern, Germany 134 G. Steidl

Fig. 1 Two frames of a traffic scene and optical flow estimation by a first order method similar to [24]. (Image source: http://ccv.wordpress.fos.auckland.ac.nz/eisats/set-7/) non-smooth, convex functionals with generalized total variation regularization terms which include second order derivatives. Let us first have a look at the objects we are dealing with.

1.1 Digital Images

A digital two-dimensional gray-value image f can be considered as a mapping from a two-dimensional grid G := {1,...,m}×{1,...,n} into a discrete subset {0,..., L − 1} of N, whose values can be visualized as gray values by a computer screen. Humans can distinguish between 64 gray values; however often 8-bit images with L = 256 different values are considered. When dealing with images we usually assume that f maps into the real numbers (or an interval) instead of {0,...,L− 1} and apply a subsequent quantization (rounding). In Sect. 4 we present a typical discrete approach to a variational problem via finite differences which relies on this image model. From the analytical point of view it is often more intuitive to consider gray-value images as mappings from a bounded domain Ω ⊂ R2 into the real numbers. Then the functionals have to be minimized over certain (infinite dimensional) function spaces. (It is not clear to which function spaces natural images really belong, see [56].) For the numerical treatment one has finally to apply a discretization, but not necessary a finite difference approach. Here we refer to the books [38, 85] which focus on continuous models. In Sects. 2 and 3 we start with a continuous variational approach which may be useful for people having some background in function spaces. It is possible to skip these sections and move directly to the discrete, finite dimensional setting on G in Sect. 4. Gray-value images are the “simplest” digital images. Often we have to deal with vector-valued functions f : G → Rd . For two-dimensional flow fields appearing, e.g., in motion estimation we have d = 2. A typical example is shown in Fig. 1. Based on the estimation of optical flow fields we will estimate strain tensors in Sect. 5.For hyperspectral images d is rather large, e.g., d = 256. For color images on a computer screen we have usually d = 3 and the three components represent the colors red (R), green (G) and blue (B) (again each color is in {0,...,L− 1}, but for simplicity we assume that we have real values). Depending on the visualization device other color Combined First and Second Order Variational Approaches for Image Processing 135

Fig. 2 Color representation systems: RGB cube and HSV cone. (Image sources: http://commons. wikimedia.org/wiki/File:RGB_color_cube.svg and http://commons.wikimedia.org/wiki/File:HSV_cone. jpg)

Fig. 3 Top: Original image sailboat and its hue. Bottom: Its saturation and value of brightness. (Image source: USC-SIPI Image Database) spaces than RGB are used. One example is the HSV space which splits the image into its hue, saturation and value of brightness. The hue describes in each area of an image the dominant color ingredient that one really perceives. It is a cyclic value between 0 (for red) and 2π, see Fig. 2. While it is quite hard to determine a certain color by just addressing its RGB components, the HSV space is more intuitive to the 136 G. Steidl human color reception system. Figure 3 shows the decomposition of an image into its HSV components. In variational models with vector-valued images, the meaningful coupling of the different vector components is an interesting ongoing research topic, see [52, 81]. We cannot address this circumvent topic in the present paper. Finally, in various applications we have to replace Rd by a d-fold Riemannian manifold Md . Typical examples are the manifold of symmetric positive deﬁnite matrices which appears in diffusion tensor MRI, the rotation group SO(3) or the n-sphere Sn. Often cyclic data have to be handled which is the topic of Sect. 6.An example of an S1-valued image is the hue image in Fig. 3. Although the S1 is a rather simple manifold the new techniques from convex optimization recently used to minimize the variational functionals do not simply carry over to this setting.

1.2 Variational Models

In image restoration we are interested in ﬁnding an unknown image u0 given its modiﬁed version f . Mathematically the corruption of the original image can be often described by the application of a linear operator K to u0. For example, for blurred images, K is a convolution operator, and for images with missing areas a masking operator, see Fig. 11. The restoration of missing pixels is called “inpainting”. In addition the result of the linear transform Ku0 is often corrupted by noise. Depending on the application we have to tackle with different kind of noise. In this paper we simply focus on additive white Gaussian noise. In summary, the observed image is

f = Ku0 + η, where η is the realization of a Gaussian independently identically distributed random vector with mean zero and variance σ . In pure denoising problems K is just the identity operator. Often K is known which is also assumed in this paper. If K is also unknown, the reconstruction problem becomes much harder. The deblurring problem with an unknown convolution kernel is called “blind deblurring”. The problem to reconstruct u0 from f ≈ Ku0 is ill-posed or ill-conditioned, since the operator K is either not invertible as for the inpainting problem or has no continuous inverse (resp. is ill-conditioned in the finite dimensional setting) as in the deblurring problem such that small errors in f are amplified if we just apply the inverse operator to f . Problems of this kind were traditionally considered in the field of inverse problems, see [14, 44, 66]. A general approach is to minimize an appropriate functional of the form D(Ku; f)+ αR(u) → min, (1) u and to take a minimizer as approximation of u0. The so-called data fidelity term D measures the deviation of Ku from f . It is often smooth and convex and becomes minimal if Ku = f . The regularizing term R is required to make the problem well- posed. Moreover, it contains a priori knowledge (desires) on the reconstructed images and is also called image prior. The design of appropriate regularizers such that the minimizers of (1) have certain properties like the preservation of important image structures while keeping the functional analytically and numerically feasible is a Combined First and Second Order Variational Approaches for Image Processing 137 challenging task. Since such goals cannot be achieved with differentiable regularization terms, current practice almost exclusively uses non-smooth regularization functionals like the total variation (TV) [79] and generalizations thereof, or 1-norms of coefficients arising from scalar products with frame systems, see, e.g., [48, 73]. The analysis of such functionals requires tools from functional analysis as well as convex analysis. Finally, the regularization parameter α balances the influence of the data and the regularization term. Its appropriate choice is very important and an ongoing topic of research. Several techniques were developed to address this task as Morozov’s dis- crepancy principle [75], the L-curve criterion [68], the generalized cross-validation [98], normalized cumulative or residual periodogram approaches [80], and variational Bayes’ approaches [4, 57]. For an overview over the first methods we refer to [58] and for recent relations between penalized and constrained problems to [95] and the references therein. In this paper we do not pay attention to an automatic adjustment of the parameters but tune them by hand according to some error measures.

1.3 Algorithms

The efficient solution of non-smooth variational problems of the form (1) demands for appropriate algorithms. Taking into account the specific structure of the functional as a sum of different convex terms, splitting algorithms are a quite canonical choice. Their strength consists in the splitting of the original problem into a sequence of smaller ones which are easy to compute. Operator splitting methods were first applied to the numerical solution of partial differential equations in the 60th of the last century, e.g., by Douglas and Rachford 1956 [43]. Here the splitting was applied to linear, single-valued operators in order to solve a certain linear system of equations efficiently. More than 20 years later these splitting methods were generalized in the convex analysis community to the solution of inclusion problems, where the linear operators have to be replaced by nonlinear, set-valued, monotone operators. Here we refer to the initial paper of Li- ons and Mercier [71] which was followed by many others. Again after more than 20 years these methods became popular in image processing starting with the work of Combettes and Wajs [37]. Operator splittings in conjunction with (augmented) La- grangian methods and primal-dual methods have recently received a lot of interest, for an overview see, e.g., [25, 36]. Interestingly, one of the first papers in imaging in this direction by Goldstein and Osher [53] proposes a so-called alternating split Bregman algorithm. Later it turned out that from one point of view this algorithm coincides with the Douglas–Rachford algorithms applied to the dual problem and from another point of view with the alternating direction method of multipliers based on the augmented Lagrangian, see [45, 86]. In this paper we will show how primal-dual methods can be applied to minimize appropriate functionals which were proposed in two recent applications, namely in the computation of local deformations in materials from a sequence of image frames via optical flow estimation [7], and the denoising of phase-valued images [10]. 138 G. Steidl

2 Continuous First Order Variational Models

2 Let Ω ⊂ R be a bounded Lipschitz domain. Assume that f ∈ L2(Ω) was obtained by applying a linear operator K, e.g., a convolution operator, to an unknown function u0 ∈ L2(Ω) and the result was corrupted by additive white Gaussian noise. Then the log likelihood approach leads to the minimization of the data fidelity term 1 D(Ku; f):= f − Ku2 . (2) 2 L2 For other noise statistics the data term must be replaced by appropriate ones as the Kullback–Leibler divergence in case of Poisson or certain kind of multiplicative noise, see, e.g., [82, 89]ortheL1 norm for impulse noise [26]. For the choice of the regularizer, the W 1,2 semi-norm R(u) := ∇u2 has a long tradition: L2 1 f − Ku2 + α |∇u|2 dx, α > 0. L2 2 Ω However, this model is not appropriate for most image processing task, simply since the minimizer is too smooth: From the continuous point of view the minimizing image must be in the “smooth” Sobolev space W 1,2. In particular, for the identity operator K, the gradient flow of the corresponding Euler–Lagrange equation is the linear partial differential equation ut = αu − u + f . Hence the minimizer is a uniformly smoothed image which does not take prominent structures of f like edges into account. In summary, it is impossible to reconstruct sharp edges with semi-norms of W 1,p, p>1. This is demonstrated in Fig. 4. This drawback leads to the consideration of new classes of non-smooth regularizers, where the TV regularization [79] is certainly the most frequently applied one in variational image processing. Let BV(Ω) denote the space of functions of bounded variation, i.e., the Banach space of functions u : Ω → R with finite norm := + u BV u L1 TV(u), where := : ∈ 1 R2 ≤ TV(u) sup u div ϕdx ϕ Cc Ω, , ϕ ∞ 1 , Ω

1 := 2 + 2 2 1 R2 and ϕ ∞ (ϕ1 ϕ2 ) L∞ .HereCc (Ω, ) denotes the space of continuously differentiable vector-valued functions on Ω with compact support. The distributional ﬁrst order derivative Du of u is a vector-valued Radon measure with total variation |Du|(Ω) = TV(u). In particular, for u ∈ W 1,1(Ω) the TV term reduces to TV(u) = |∇ | Ω u dx so that our functional becomes 1 f − Ku2 + α |∇u| dx, α > 0. L2 2 Ω Since the gradient of a constant area is zero, the functional does not penalize constant areas which leads to well-known staircasing effect. A good overview on total variation for image analysis in particular the relation between the TV functional and the perimeter of level sets can be found in [27]. A typical example can be seen in Fig. 4. Combined First and Second Order Variational Approaches for Image Processing 139

Fig. 4 Top: Original image and noisy image. Bottom: restored images with W 1,2 regularization and with TV regularization. The W 1,2 regularized method produces a smoothed images where important image structures get lost. The TV regularized approach keeps sharp edges, but introduces a staircasing effect in non ﬂat areas. (Image from [41])

We mention that smoothed versions of the absolute value function like its Moreau en- := 1 ·− 2 +|·| := |·|2 + 2 velope also known as Huber function ψ mint 2ε ( t) or ψ ε can reduce the staircasing artifact to some extent but have other drawbacks.

3 Continuous Second Order Variational Models

A successful method to cope with the staircasing effect is to incorporate higher order derivatives into the regularization term, in particular second order derivatives. For simplicity we incorporate the regularization parameters within the regularizer R in the rest of the paper. 2 Let BV(Ω, R ) denote the space of vector-valued functions u = (u1,u2) of bounded variation, where the above definition of TV has to be modified by + := Ω u1 div ϕ1 u2 div ϕ2 dx. We consider the space of bounded Hessians BH(Ω) 1,1 2 2 {u ∈ W (Ω) :∇u ∈ BV(Ω, R )} with norm uBH =uBV +|D u|(Ω) and |D2u|=|D(∇u)|,see[2, 40]. There are three established ways for combining first and second order derivatives in the regularization term: (i) The first approach minimizes the functional (1) with data term (2) and regularization term R(u) := α|∇u| dx + βD2u(Ω) Ω over BV(Ω, R2). For applications in imaging we refer to [76]. 140 G. Steidl

(ii) The second method makes use of the inﬁmal convolution of two (extended) functionals Ri , i = 1, 2 which is deﬁned by

(R1R2)(u) := inf R1(v) + R2(w). v+w=u The inﬁmal convolution is the counterpart of the ‘usual’ convolution of functions in the max-plus algebra. More precisely, the regularization term consists of the 2 inﬁmal convolution of R1(v) := TV(v) and R2(w) := |D w| for u (and v)in BV(Ω) and w ∈ W 1,1 with ∇w ∈ BV(Ω, R2): R(u) := inf αTV(v) + βD2w(Ω). v+w=u

For ∇v ∈ W 1,1(Ω, R2) and u ∈ W 1,1(Ω) the whole functional can be rewritten as 1 f − Ku2 + α|∇u −∇w| dx + β∇2w dx → min (3) L2 2 Ω u,w Roughly speaking, for K = Id, we ‘first’ approximate locally the gradient of the function f by ∇w that has itself a low total variation and then find u as an approximation of f such that u − w has a low total variation. As a consequence we do not get anymore an almost piecewise constant minimizer. This regularization method was introduced in imaging by Chambolle and Lions [28]. For a recent generalization called ICTV we refer to [63]. (iii) A conceptually different approach was recently proposed by Bredies, Kunisch and Pock [21]; for the discrete setting see also [87]. Here the regularization term is the so-called total generalized variation (TGV2) defined by R = 2 (u) TGV(α,β)(u) := 2 : ∈ 2 R2 sup u div ϕdx ϕ Cc Ω,Sym , Ω ϕ∞ ≤ α, div ϕ∞ ≤ β

2 R2 where Cc (Ω, Sym( )) denotes the space of two times continuously differentiable 2-tensor ﬁelds with compact support in Ω and divk ϕ := tr(∇k ⊗ ϕ), k = 1, 2. In other words, we have for ϕ = (ϕ1,1,ϕ1,2,ϕ2,2) that

2 div ϕ = ∂xxϕ1,1 + (∂xy + ∂yx)ϕ1,2 + ∂yyϕ2,2

and

div ϕ(e1) = ∂xϕ1,1 + ∂yϕ1,2, div ϕ(e2) = ∂xϕ1,2 + ∂yϕ2,2.

2 An alternative deﬁnition of TGV(α,β) is given by 2 := | − | + |E | TGV(α,β)(u) min α Du y (Ω) β y (Ω), y∈BD(Ω) Combined First and Second Order Variational Approaches for Image Processing 141

where BD(Ω) is the space of functions of bounded deformation and Ey is the distributional symmetrised gradient of y,see[22]. In a simplified version, instead of (3) we get 1 f − Ku2 + α|∇u − y| dx + β|∇y| dx → min L2 2 Ω u,y There are several other approaches which make use of higher order derivatives including weighted versions of the Laplacian [33, 74], bounded Hessians (without first order terms) [13, 31, 60, 62, 72, 84], and Euler’s elastica [32, 93]. In [69] the spectral norm of the discrete Hessian is used as a regularizer. The relation to higher order diffusion filters was analyzed by Didas et al. in [42]. For discrete versions of the above approaches we refer to [87, 88].

4 Discrete Second Order Variational Models

For computations we need a discrete version of the functionals in the previous section. We consider gray-value images as functions f : G → R defined on a rectangular grid G := {1,...,m}×{1,...,n} and make use of finite difference approximations for the differential operators. Indeed the appropriate discretization plays an important role in certain imaging applications, see, e.g., [100, 102]. In particular, if the second order derivatives come into the play and the identities satisfied by the continuous operators should be preserved it becomes necessary to work on more than one grid. One possibility is the use of the finite mimetic difference method, see, e.g., [65, 103, 104]. In the following we will use simple forward difference operators. To have a con- venient vector-matrix notion it is usual to reorder f columnwise into a vector vec f of length N := mn. If the meaning is clear from the context we keep the notation f instead of vec f . The discretization of the data term (2) with a linear operator K follows just by considering the functions on the grid and taking the Euclidean norm of their reshaped form: 1 D(Ku; f):= f − Ku2,K∈ RN,N. 2 2 For discretizing the regularization term, let ⎛ ⎞ −11 ⎜ . . ⎟ ⎜ .. .. ⎟ n,n Dn := ⎜ ⎟ ∈ R ⎝ −11⎠ 0 be the discrete forward difference operator with respect to mirror (Neumann) boundary conditions and Dx ∇:= with Dx := In ⊗ Dm,Dy := Dn ⊗ Im Dy 142 G. Steidl be the discrete gradient operator. Here ⊗ denotes the Kronecker product operator. Further, let ⎛ ⎞ ⎛ ⎞ T Dxx Dx 0 2 ⎝ ⎠ ˜ ⎝ T T ⎠ ∇ := Dxy + Dyx , ∇:= Dy Dx T Dyy 0 Dy be a second order difference operator and the symmetrised discrete gradient operator of a vector field, resp., where

:= − ⊗ T := − T ⊗ Dxx In DmDm,Dyy DnDn Im, := − T ⊗ := − ⊗ T Dxy Dn Dm,Dyx Dn Dm. Then the regularization terms in (i)–(iii) become 2 (i) R(u) = α∇u2,1 + β∇ u2,1, R = ∇ − + ∇2 (ii) (u) minw∈RN α (u w) 2,1 β w 2,1, R = ∇ − + ∇˜ (iii) (u) miny∈R2N α u y 2,1 β y 2,1, d N Nd where we use the mixed L2(R ) − L1(R ) norms deﬁned for x ∈ R as

1 N d−1 2 2 x d N =x := x , L2(R ),L1(R ) 2,1 jN+i i=1 j=0 see [87, 88] for the functionals (ii) and (iii). Figure 5 shows the denoising results for the image in Fig. 4 top, right. There are numerous applications of second order models in image processing. In the following we present two recent ones, namely, the computation of local deformations in materials from a sequence of image frames via optical flow estimation [7], and the denoising of phase-valued images [10]. The first application is interesting since the strain is directly computed within the numerical algorithm by an appropriate variable splitting. The second application establishes for the first time a second order model for cyclic data.

5 Computation of Local Deformation for Multiphase Metallic Materials

5.1 Motivation

The (Cauchy) strain tensor plays a fundamental role in mechanical engineering for deriving local mechanical properties of materials. In this subsection we calculate local strains of silicon carbide particle reinforced aluminum matrix composites from a sequence of scanning electron microscope images acquired during tensile tests, where the micro-specimen is pulled in horizontal direction and elongates with increasing force. Figure 6 illustrates the experimental setup and the resulting image sequence schematically.1 For deformations of a continuum body the strain tensor is

1The scanning electron microscope tensile tests were performed at the Department of Mechanical and Process Engineering, University of Kaiserslautern by Dr. F. Balle, Prof. Dr. D. Eiﬂer, and S. Schuff. Combined First and Second Order Variational Approaches for Image Processing 143

Fig. 5 Top: Noisy image from Fig. 4 and denoised images with regularization (i). Bottom: Denoised images with regularization (ii), and (iii)

Fig. 6 Experimental setup for the tensile test inside a scanning electron microscope. Load-deformation diagram with three selected micrographs (1024 × 884 pixel; size 51 µm × 44 µm) taken under increasing load. (Image credit: S. Schuff, Department of Mechanical and Process Engineering, University of Kaiser- slautern) 144 G. Steidl deﬁned via the gradient of the displacement u = (u1,u2), i.e., ε ε 1 ∂ u 1 (∂ u + ∂ u ) ε = xx xy := ∇uT +∇u = x 1 2 y 1 x 2 . ε ε 1 + yx yy 2 2 (∂yu1 ∂xu2)∂yu2

We will especially focus on εxx which describes the change in displacement u1 for the horizontal direction. A positive value indicates tension and a negative one com- pression. One of the few papers which addresses the (Lagrangian) strain tensor computation by a variational method is [59] by Hewer et al. The authors propose a smooth fourth order optical flow model which directly computes the strain tensor from an image sequence obtained in a biaxial tensile test with an elastomer. In contrast to our paper they were interested in the macro scale behavior and compute the minimizer of their smooth energy function by solving the corresponding Euler–Lagrange equations. For engineering purposes there are some approaches and commercial software packages based on image correlation such as [19, 83, 94, 99]. However, these methods are only suitable for computing the strain on a macro scale. Variational methods for optical flow estimation go back to [64]. There is a vast number of refinements and extensions of their approach and we refer to [9, 61] for a comprehensive overview. In particular higher order optical flow models were successfully used, e.g., in [1, 96, 103–105].

5.2 Variational Model

2 As for images we consider flow fields u = (u1,u2): G → R on a rectangular grid T T T 2N and use the reordering vec u = ((vec u1) ,(vec u2) ) ∈ R . As usual we identify u with vec u if the meaning is clear from the context. In the task at hand we focus on two images f1 and f2 from the tensile test belong- ing to different tensions. For multiframe treatments in optical flow we refer, e.g., to [99]. The data term in (1) relies on an invariance requirement between these images. Here we focus on the brightness invariance assumption which reads in the continuous setting as f1(x, y) − f2 (x, y) − u(x,y) ≈ 0,u= (u1,u2). (4)

Using a first order Taylor expansion around an initial optical flow field u¯ = (u¯1, u¯2) gives ∂xf2 f2 (x, y) − u ≈ f2 (x, y) −¯u − (x, y) −¯u · u(x,y) −¯u(x,y) . (5) ∂yf2

Later we will apply a coarse-to-ﬁne scheme [3, 23] and use the result from one scale as an initialization for the next scale. By (5) the requirement (4) becomes ∂xf2 0 ≈ f1(x, y) − f2 (x, y) −¯u + (x, y) −¯u · u(x,y) −¯u(x,y) . ∂yf2

Note that f2((x, y) −¯u) is only well deﬁned in the discrete setting, if (i, j) −¯u is in G. Here, bilinear interpolation is used to compute values between grid points. Combined First and Second Order Variational Approaches for Image Processing 145

Using a non-negative increasing function ϕ : R → R≥0 and considering only grid points (x, y) = (i, j) ∈ G, the data term becomes −¯ Dmf2((i, j) u) · −¯ − −¯ + ϕ −¯ T u(i, j) u(i, j) f2 (i, j) u f1(i, j) f2((i, j) u)Dn (i,j)∈G and for ϕ(t) := |t| ﬁnally 1 D(Ku; c) := Ku+ c ,K:= (AB), (6) 2 1

2N where ·1 denotes the L1(R ) norm and := −¯ := −¯ T A diag vec Dmf2 (i, j) u ,Bdiag vec f2 (i, j) u Dn , −¯ := − Dmf2((i, j) u) ·¯ − −¯ + c vec −¯ T u(i, j) f2 (i, j) u f1(i, j) . f2((i, j) u)Dn For the choice of the regularization term we notice that the global displacement during insitu tensile testing can be roughly approximated by a plane. Therefore it makes sense to separate this global displacement from the local one using an inﬁmal convolution regularization (ii) (or the TGV2 model which provides nearly the same results): 2 R(u) = inf α∇v2,1 + β ∇ w , (7) v+w=u 2,1

2 2 where ∇u := (I2 ⊗∇)u and ∇ u := (I2 ⊗∇ )u. It can be shown that in the case ker( AB) ∩ ker(∇) ={0} there exists a minimizer of J(u):= D(Ku; c) + R(u),see[7].

5.3 Algorithm

To solve the convex, non-smooth optimization problem (1) with data term (6) and regularization term (7) we apply a primal dual algorithm with an appropriate splitting. To this end, we rewrite the problem as min Ku+ c1 + αs2,1 + βt2,1 u,w,s,t (8) ∇ −∇ u s such that = . 0 ∇2 w t

Indeed, we are not interested in u, but in its non-smooth part v = u − w or more precisely in the strain s = ε(v) =∇(u − w) which fortunately directly appears in the constraint. With F(u,w) := Ku + c1 and G(s, t) := αs2,1 + βt2,1 the Lagrangian of (8) reads L (u,w,s,t),(p,q) := F(u,w)+ G(s, t) + ∇(u − w) − s,p + ∇2w − t,q 146 G. Steidl with the dual variable (p, q). Then the primal problem (8) and its dual can be written as (P ) min max L (u,w,s,t),(p,q) , u,w,s,t p,q (D) max min L (u,w,s,t),(p,q) . p,q u,w,s,t Using the Fenchel dual of G deﬁned by ∗ s p G (p, q) := max , − G(s, t) s,t t q the problems become ∗ (P ) min max F(u,w)− G (p, q) + ∇(u − w),p + ∇2w,q , u,w p,q ∗ (D) max min F(u,w)− G (p, q) + ∇(u − w),p + ∇2w,q . p,q u,w The Arrow–Hurwitz method consists in alternating the minimization over the primal and dual variables and adding a stabilizing quadratic terms: (r+1) ∇ − (r) u = + (u w) p (r+1) arg min F(u,w) 2 , (r) w u,w ∇ w q (r) 2 + 1 u − u (r) , 2τ w w 2 u(r) ∇Tp(r) = prox − τ τF w(r) −∇Tp(r) + (∇2)Tq(r) (r+1) ∇ (r+1) − (r+1) p = ∗ − (u w ) (r+1) arg min G (p, q) 2 (r+1) q p,q (∇ )w (r) 2 + 1 p − p (r) 2σ q q 2 p(r) ∇(u(r+1) − w(r+1)) = prox ∗ + σ , σG q(r) (∇2)w(r+1) where the proximal mapping of a proper, closed convex function g : Rd → Rd ∪ {+∞} is deﬁned by

:= 1 − 2 + proxλg (x) arg min x y 2 λg(y), λ > 0. (9) y 2 The internal parameters τ and σ of the algorithm can be found as in [54]. The basic idea behind this algorithm is that the proximal mappings can be simply computed (often there exists an analytical formula) for many functions used in variational image reconstruction models including the present ones. Combined First and Second Order Variational Approaches for Image Processing 147

Fig. 7 Results corresponding to images under low and high load. Zooms into two different regions. Top: The two initial image regions without load. Bottom: εxx = ∂x v1 for the two different regions under low and high load together with the image frame. The cracks in the high load images correspond to areas of high strain εxx. The high strain areas can also be detected from the low load images, where the cracks in the images themselves are (nearly) not visible

Further, we apply an extrapolation step for the dual variable. This results in the primal-dual algorithm [29, 77] which is known as Chambolle–Pock algorithm in image processing. It was meanwhile generalized for various other functionals, e.g., in [20, 35, 97]. To cope with large displacements our algorithm was moreover updated by a coarse-to-ﬁne scheme, see, e.g., [3, 23].

5.4 Numerical Results

In this tutorial we demonstrate the potential of our variational approach only by one example. We have applied our method to two different images f2 obtained during the tensile test at a low (F1 = 3446N) and a high (F2 = 3980N) load level, see Fig. 6. Figure 7 shows magniﬁed micrographs (100 × 100 pixels, size 5 µm × 5µm)oftwo different regions of the whole image where cracks were initiated. In both experiments the areas of computed high local strains correspond to crack areas in the material. It is remarkable that even under low load, when the cracks are not or hardly visible in the images, the strain tensor in the corresponding regions is high and seems to be a sensitive and useful tool to study crack initiation mechanisms of silicon carbide reinforced aluminum matrix composites. For details on the parameter choice see [7]. 148 G. Steidl

Fig. 8 Left to right: Image sailboat from Fig. 3 with hue channel corrupted by wrapped Gaussian noise (σ = 0.4), denoised image by a second order method for real valued data (wrong colors are created), and by our method for cyclic data. Without taking the cyclic structure of the data into account we do not obtain correct colors

6 Restoration of Cyclic Data

6.1 Motivation

In various applications in image processing and computer vision the functions of interest take values on the circle S1 or on manifolds. In this subsection we deal with S1-valued data which appear, e.g., in interferometric synthetic aperture radar [17, 18, 39], electroencephalogram data analysis [90], ground based astronomy [8, 30, 34, 55], circular statistics [47, 67] and color image restoration in HSV spaces. Handling phase-valued data with techniques for real-valued data introduces artifacts since the cyclic nature of the data is neglected. Figure 8 shows a denoising example, where the so-called hue of an image which is phase-valued was denoised by minimizing the functional with squared L2 data term and second order regularizer (i) described in Sect. 4. Without taking the cyclic structure of the data into account the color values became completely wrong. Although the minimization of TV regularized functionals is among the most popular methods for edge-preserving image restoration such methods were only very recently applied to cyclic structures. The TV notation for functions with values on a manifold has been studied in [50, 51] using the theory of Cartesian currents. These papers were an extension of the previous work [49] were the authors focus on S1-valued functions and show in particular the existence of minimizers of certain energies in BV(Ω). The ﬁrst papers which apply a cyclic TV approach in imaging are [70, 91, 92]. In this subsection we want to incorporate second order differences into the energy functional for cyclic data. Indeed [10] is the ﬁrst paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration.

6.2 Variational Model

We restrict our attention to functionals (1) with K = Id. First we replace the 1 Euclidean distance by the geodesic distance dS1 (p, q) := arccos(p,q) on S := { 2 + 2 = : = T ∈ R2} ∈ S1 p1 p2 1 p (p1,p2) . Given a base point q , the exponential Combined First and Second Order Variational Approaches for Image Processing 149

: R → S1 S1 R S1 S1 map expq from the tangent space Tq of at q onto is 2π- = ∈ R periodic, i.e., expq (x) expq ((x)2π ) for any x , where (x)2π denotes the unique point in [−π,π) such that x = 2πk + (x)2π , k ∈ Z. To guarantee the injectivity of the exponential map, we restrict its domain of deﬁnition from R to [−π,π). Thus, ∈ S1 ∈[− = for p,q , there is now a unique x π,π) satisfying expq (x) p. Given such 1 representation system xj ∈[−π,π) of pj ∈ S , j = 1, 2 centered at an arbitrary point q on S1 the geodesic distance becomes dS1 (p1,p2) = d(x1,x2) = min|x2 − x1 + 2πk|= (x2 − x1)2π . (10) k∈Z

Then, for f,u : G → S1, the cyclic equivalent of the discretized data term in (2) becomes m,n 1 2 D(u; f):= d(ui,j ,fi,j ) . 2 i,j=1 To define a regularization term related to those in (i) in Sect. 4 we need appropriate definitions of first and second order difference operators for cyclic data. For w = κ ∈ Rκ \{ } κ = (wj )j=1 0 with j=1 wj 0 a real-valued finite difference operator is given by x → x,w. For cyclic data we consider similarly d(x; w) := min [x + α1κ ]2π ,w , (11) α∈R where 1κ is the vector with κ entries 1 and [x]2π denotes the component-by- component application of (t)2π if t = (2k + 1)π, k ∈ Z and [(2k + 1)π]2π =±π, T k ∈ Z.Forw1 := (−1, 1) definition (11) coincides with the geodesic distance, i.e., 2 T d(x; w1) = d(x1,x2) for all x ∈[−π,π) . For the difference masks w2 := (1, −2, 1) T and w1,1 := (−1, 1, 1, −1) it can be shown [10] that ; = ∈ d(x wν) x,wν 2π ,ν 2,(1, 1) . It is remarkable that this relation does not hold true for other differences, e.g., for the third order difference w = (−1, 3, −3, 1)T. Introducing the notation

d2(x) := d(x; w2) and d1,1(x) := d(x; w1,1) the regularization term becomes R := + hv + d (u) TV1(u) TV2 (u) TV2(u) (12) with n−1,m n,m−1 TV1(u) := α1 d(ui,j ,ui+1,j ) + α2 d(ui,j ,ui,j+1), i,j=1 i,j=1 n−1,m n,m−1 hv := + TV2 (u) β1 d2(ui−1,j ,ui,j ,ui+1,j ) β2 d2(ui,j−1,ui,j ,ui,j+1), i=1,j=2 i=2,j=1 150 G. Steidl

n−1,m−1 d := TV2(u) β3 d1,1(ui,j ,ui+1,j ,ui,j+1,ui+1,j+1). i,j=1 Note that this regularizer corresponds to the so-called “anisotropic counterpart” of the regularization term in (i) in Sects. 3 and 4, i.e., roughly speaking it is related to | |+ | |+ | |+ | |+ | | α1 ∂x1 u α2 ∂x2 u β1 ∂x1x1 u β1 ∂x2x2 u β3 ∂x1x2 u dx. Ω 6.3 Algorithm

As in the previous section we will apply an iterative algorithm which is based on analytic computations of proximal mappings of the difference functions d(·; w) similar to (9).

Proximal Mapping of Differences of Cyclic Data As a cyclic counterpart to (9) we deal with := E ; proxλd(·;w)(f ) arg min (x g,w), x∈[−π,π)κ κ 1 2 p E(x; g,w) := d(xj ,fj ) + λd(x; w) ,λ>0 2 j=1 ∈{ } for w w1,w2,w1,1 .Hereargminx∈[−π,π)κ means that we are looking for the rep- resentative of x ∈ (S1)κ in [−π,π)κ .

κ Theorem 6.1 [10] For w ∈{w1,w2,w1,1} set s := sgn(f,w)2π . Let g ∈[−π,π) , where κ is adapted to the respective length of w.

1. If |(g,w)2π | <π, then the unique minimizer of E(x; g,w) is given by |(g,w) | xˆ = (g − smw) ,m:= min λ, 2π . 2π 2 w 2

2. If |(g,w)2π |=π, then E(x; g,w) has the two minimizers π xˆ = (g ∓ smw) ,m:= min λ, . 2π 2 w 2 Our proof in [10] uses the fact that κ 1 2 E(x; f,w) = min|gj − xj − 2πkj | + λ min x,w−2πσ 2 kj ∈Z σ ∈Z j=1 = 1 − − 2 + − min g x 2πk 2 λ x,w 2πσ , k∈Zκ σ∈Z 2 so that the minimization can ﬁnally be restricted to those of real-valued data. Combined First and Second Order Variational Approaches for Image Processing 151

Cyclic Proximal Point Algorithm (CPPA) Having an analytic expression for the proximal mappings of the cyclic finite differences at hand, our algorithm of choice to minimize the whole functional J(u):= D(u; f)+ R(u) with D(u; f) in (10) and R(u) in (12) will be the cyclic proximal point algorithm (CPPA). The basic idea of the CPPA consists again in splitting the functional in an = L appropriate way as a sum J l=1 Jl and to compute successively the proximal mappings of the summands Jl. The proximal point algorithm (PP) on the Euclidean space goes back to [78] and was extended to Riemannian manifolds of non-positive sectional curvature in [46] and to Hadamard spaces in [5]. Its real-valued version computes arg minx g(x) for a proper, closed convex function g on RN by iterating (r) := (r−1) = 1 (r−1) − 2 + x proxλg x arg min x x 2 λg(x). x 2 + Finding arg minx g1(x) g2(x) for the sum of two proper, closed convex function gi , i = 1, 2 could be at the first glance performed similarly x(r) := x(r−1) . proxλg2 proxλg1 Since the concatenation of the two firmly non-expansive proximal operators is again (r) firmly non-expansive the sequence of iterates {x }r converges, but unfortunately not λ λ to a minimizer of g1 + g2 but instead to the minimizer of g1 + g2, where g1 denotes the Moreau envelope of g1. Convergence to the minimizer of the desired sum g1 + g2 can by achieved by varying the parameter λ within the iterations. More precisely, the L CPPA computes arg minx gl(x) by l=1 x(k) = ... x(k−1) , proxλkgL proxλkgL−1 proxλkg1 where ∞ ∞ =∞ 2 ∞ λk , and λk < . (13) k=0 k=0 This CPPA on the Euclidean space was given in [16], see also the survey [15]. A CPPA for Hadamard spaces can be found in [6]. With first order differences of cyclic data in the regularization term the CPPA was applied for image restoration in [101]. For our functional J we use a splitting into L = 6 (in one dimension) and L = 15 (in two dimensions) summands which correspond to the natural splitting of the image grid into disjoint parts by the difference operators. For example we apply for the first hv summand in TV2 the splitting into the three parts

n−1 2 3 m d2(u3i−2+ν,j ,u3i−1+ν,j ,u3i+ν,j ). ν=0 i=1 j=1 152 G. Steidl

Convergence Since S1 is not a Hadamard space, the convergence analysis of the CPPA in [6] cannot be applied. However, convergence can be proved under certain conditions: (i) the data f ∈ (S1)n,m is locally dense enough, which means that the distance between neighboring pixel values

d∞(f ) := max max d(fi,j ,fk,l), (i,j)∈I (k,l)∈Ni,j Ni,j := (k, l) ∈ I :|i − k|+|l − j|=1 is sufﬁciently small. (ii) The regularization parameters α, β are sufﬁciently small. More precisely, we assume that for some ε>0wehave

ε2 TV (f ) + TVhv(f ) + TVd(f ) ≤ , (14) 1 2 2 μ where μ := max{α1,α2,β1,β2,β3} > 0. (iii) Condition (13) must be fulﬁlled.

∈ S1 n,m π := { } Theorem 6.2 [10] Let f ( ) with d∞(f ) < 8 . Let λ λk k fulﬁll property (13) and π ε2 + 32α2L(L + 1) + 8α∞L< , 2 16 for some ε>0, where L = 15. Further, assume that the parameters α, β of the func- (k) tional J and ε satisfy (14). Then the sequence {u }k generated by the CPPA converges to a global minimizer of J .

6.4 Numerical Results

In this section we demonstrate the performance of our algorithms by numerical examples with artificial signals and images. For real-world applications in synthetic aperture radar and for the analysis of electroencephalograms we refer to [10]. We start by demonstrating the influence of first and/or second order differences on the denoising process for one-dimensional data. The function g :[0, 1]→[−π,π) given by ⎧ ⎪−24πx2 + 3 π for 0 ≤ x ≤ 1 , ⎪ 4 4 ⎪ ⎪ πx − π 1

Fig. 9 Denoising of a one-dimensional signal with CPPA. (a) Original signal u0 (dashed red) and dis- turbed signal by wrapped Gaussian noise f (solid black). (b) Shows the signals from (a) with respect to −3 [0, 2π).(c)–(e) Reconstructed signals ur using (c) only the TV1 regularizer, e(u0,ur) ≈ 6.06 × 10 , −3 −3 (d) only the TV2 regularizer, e(u0,ur) ≈ 4.34 × 10 ,and(e) both of them, e(u0,ur) ≈ 3.53 × 10 . While (c) suffers from the staircasing effect, (d) shows weak results at constant areas. The combination of both regularizers in (e) yields the best image. (f) Shows signal (e)in[0, 2π)

− 5 [− change from π to π at 16 is just due to the chosen representation system in π,π). − 7 Similarly the two constant parts with the values π and 8 π differ only by a jump − π [ size of 8 . In the representation system 0, 2π) the signal looks as in Fig. 9(b). We apply our method with different model parameters α and β to f which yields the restored signals ur. The parameters were optimized with respect to the restoration 154 G. Steidl

Fig. 10 Denoising of an artificial surface: (c) wrapped image (b)of(a) corrupted by wrapped Gaussian noise. Reconstructed images ur using (d) the real-valued first plus second order regularization (i). The −3 results at near ±π are poor. (e) the cyclic TV1 regularizer, e(u0,ur) = 7.09 × 10 .Hereur reproduces the piecewise constant parts of the stairs in the upper left part and the background, but introduces a staircasing in both linear increasing areas inside the ellipse and in the half ellipsoid. (f) the regularizer in (12), −3 e(u0,ur) = 5.37 × 10 . Combining first and second order cyclic differences improves the results Combined First and Second Order Variational Approaches for Image Processing 155

Fig. 11 Original image (a) and its masked version (b). The first reconstruction (c) employs only first order cyclic differences and produces staircasing. Combining first and second order cyclic differences in (d)we obtain a nearly perfect reconstruction. (Image source: Courtesy of R. Bergmann, [11]) error measured by the ‘cyclic’ mean squared error

M 1 e(u ,u ) := d(u ,u )2. 0 r N 0,i r,i i=1

The resulting signal ur is depicted in Fig. 9(c)–(e). Figure 9(f) shows the result from Fig. 9(e) with respect to another representation system. Next we consider the synthetic surface given on [0, 1]2 shown in Fig. 10(a). This surface consists of two plates of height ±2π divided at the diagonal, a set of stairs in π the upper left corner in direction 3 , a linear increasing area connecting both plateaus π having the shape of an ellipse with major axis at the angle 6 , and a half ellipsoid 9 forming a dent in the lower right of the image with circular diameter of size 25 and depth 4π. The initial data is given by sampling the described surface at m = n = 256 sampling points. Then Gaussian noise is added and the data is wrapped, see Fig. 10(b) 156 G. Steidl and (c). Figure 10(d)–(f) show the denoising results with different regularization 1 N terms. The parameters were chosen from 8 such that they minimize the cyclic mean squared error. Finally, Fig. 11 shows an artiﬁcial inpainting example with cyclic second order regularization terms. The original image u0 is corrupted by a non-invertible masking operator K which gives f . Here no noise is added. The masking operator K has to be involved into the data term of the functional. For the precise description of the setting we refer to [11]. The results were further extended to S1 × Rd -valued data in [12].

Acknowledgement Funding by the Deutsche Forschungsgemeinschaft (DFG) within the RTG GrK 1932 “Stochastic Models for Innovations in the Engineering Sciences”, project area P3, and within the DFG Grant STE571/11-1 is gratefully acknowledged.

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105. Yuan, J., Schnörr, C., Steidl, G.: Convex Hodge decomposition and regularization of image ﬂows. J. Math. Imaging Vis. 33(2), 169–177 (2009)

Gabriele Steidl received her Ph.D. and Habilitation in Mathematics from the University of Rostock, Germany, in 1988 and 1991, respectively. From 1993 to 1996, she held a position as assistant professor of Applied Mathematics at the University of Darmstadt, Germany. From 1996 to 2010, she was full professor at the Department of Mathemat- ics and Computer Science, University of Mannheim, Germany. Since 2011, she is Professor at the Department of Mathematics, University of Kaiserslautern, Germany. Her research interests include applied and computational harmonic analysis and convex analysis with applications in image processing. Jahresber Dtsch Math-Ver (2015) 117:161–164 DOI 10.1365/s13291-014-0103-9 BOOK REVIEW

Jürgen Appell, Józef Banas,´ Nelson J. Merentes Díaz: “Bounded Variation and Around” De Gruyter, 2014, 476 pp.

Michel Willem

The book under review is a remarkable treatise on the theory of functions of bounded variation. This notion was introduced by Camille Jordan in 1881 (see [4]) under the name functions of limited oscillations. His goal was to linearize Dirichlet’s condition for the convergence of Fourier series (see [2]). Let us recall Jordan’s deﬁnition: “Let x1,...,xn be a series of values of x between 0 and , and y1,...,yn the corresponding values of f(x). The points x1,y1; ...; xn, yn will form a polygon. Consider the differences

y2 − y1,y3 − y2,...,yn − yn−1. We will call the sum of the positive terms of this sequence the positive oscillation of the polygon; negative oscillation is the sum of the negative terms; total oscillation is the sum of those two partial oscillations in absolute value. Let us vary the polygon; two cases may occur: 1◦ The polygon may be chosen so that its oscillations exceed every limit. 2◦ For every chosen polygon, its positive and negative oscillations will be less than some ﬁxed limits P and N . We will say in that case that F(x) is a function of limited oscillation in the interval from 0 to ; P will be its positive oscillation; N its negative oscillation; P + N its total oscillation”. Let us also recall Dirichlet’s condition: (a) u :[a,b]→R is bounded and piecewise monotone.

M. Willem (B) Université catholique de Louvain, IRMP, Chemin du Cyclotron, 2, 1348 Louvain-la-Neuve, Belgium e-mail: [email protected] 162 M. Willem

This condition is not preserved under linear combination. On the other hand, Jordan proved that (b) u :[a,b]→R is of bounded variation is equivalent to (c) u is the difference of two increasing functions. Hence property (b) is preserved under linear combinations and (a) ⇒ (b) ⇔ (c). At the end of his paper, Jordan wrote: “The functions of limited oscillations constitute a well-defined class, whose study could be of some interest” (a remarkable example for the writing of a research project). Jordan, a specialist in group theory, wrote only two papers in the field of analysis, “Sur la série de Fourier” (1881) and “Remarques sur les intégrales définies” (1892). He was motivated by his lectures at the École Polytechnique (see [3]) where he was Professeur d’Analyse from 1876 to 1912. Functions of bounded variation will play a fundamental role in the following do- mains: – convergence of Fourier series; – rectification of curves; – integration; – duality. In 1887, in his Cours d’Analyse at the École Polytechnique, Jordan proved that the length of the graph of a continuous function u :[a,b]→R is finite if and only if u is of bounded variation. This result is so natural, that Nicolas Bourbaki attributed the definition of bounded variation to the problem of rectifiability ([1], p. 252): “Ces fonctions avaient été introduites par Jordan, à propos de la rectification des courbes”. Functions of bounded variations were used by Stieltjes in 1894 to define his integral (see [6]) and in 1919, F. Riesz discovered that every continuous linear functional on C([0, 1]) is representable by a Stieltjes integral (see [5]): 1 A f(x) = f (x)dα(x). 0 This result is the original formulation of the Riesz representation theorem identifying integrals and linear forms. Definitions of variations of functions of two variables are due e.g. to Arzelá, Fréchet, Tonelli and Vitali. Tonelli’s definition is particularly interesting, since the area, in the sense of Lebesgue, of the graph of a continuous function u :[0, 1]×[0, 1]→R is finite if and only if the variation, in the sense of Tonelli, of u is finite (see [7]). Let us add one more (classical) definition. A function u ∈ L1(Ω) is of bounded variation Jürgen Appell, Józef Banas,´ Nelson J. Merentes Díaz: “Bounded Variation and Around” 163 if the first order partial derivatives of u, in the sense of distributions, are bounded measures on Ω. This concept is also related to the problem of quadrature of surfaces (see e.g. [8]). The De Giorgi perimeter of a measurable subset A of RN is the total variation of its characteristic function: N N p(A) = sup div vdx : v ∈ D R , R , v∞ ≤ 1 . A When A is an open subset of RN with a smooth compact boundary ∂A, it follows from the divergence theorem that N N p(A) = sup v · ndσ : v ∈ D R ; R , v∞ ≤ 1 = dσ. ∂A ∂A The book under review contains beautiful expositions of those topics (except the quadrature of surfaces) and of other subjects: – nonclassical BV-spaces, – absolutely continuous functions, – nonlinear composition and superposition operators.

The composition operator Ch is deﬁned by Chf(x)= h f(x) ,a≤ x ≤ b, and the superposition operator Sh by Shf(x)= h x,f(x) ,a≤ x ≤ b. Let the norm on BV([a,b]) be

f BV = f(a) + Var f ;[a,b] , where the total variation Var(f ;[a,b]) is exactly the total oscillation of Jordan. The following problems arise:

– find necessary and sufficient conditions on h under which Chf ∈ BV (resp. Sh ∈ BV) for all f ∈ BV; – find necessary and sufficient conditions on h under which the operator Ch (resp. Sh) is continuous from BV into itself.

Only one problem is completely solved. The operator Ch maps BV into itself if and only if h is locally Lipschitz on R. We recommend the book under review. The style, clear and accurate, makes it accessible to a large public of analysts. Although suitable for graduate students, it will also be useful for researchers in the ﬁeld of nonlinear analysis.

References

1. Bourbaki, N.: Éléments d’histoire des mathématiques. Hermann, Paris (1960) 2. Dirichlet, P.L.: Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre les limites données. J. Reine Angew. Math. 4, 157–169 (1829) 164 M. Willem

3. Gispert, H.: Sur les fondements de l’analyse en France (à partir de lettres inédites de G. Darboux et de l’étude des différentes éditions du Cours d’Analyse de C. Jordan. Arch. Hist. Exact Sci. 28, 37–105 (1983) 4. Jordan, C.: Sur la série de Fourier, Paris. C. R. Acad. Sci. 92, 228–230 (1881) 5. Riesz, F.: Sur les opérations fonctionnelles linéaires. C. R. Acad. Sci. Paris 149, 974–977 (1909) 6. Stieltjes, T.: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, 1–122 (1894) 7. Tonelli, L.: Sur la quadrature des surfaces. C. R. Acad. Sci. Paris 182, 1198–1200 (1926) 8. Willem, M.: Functional Analysis: Fundamentals and Applications. Springer, New York (2013) Jahresber Dtsch Math-Ver (2015) 117:165–170 DOI 10.1365/s13291-014-0105-7 BOOK REVIEW

Luis Barreira and Yakov Pesin: “Introduction to Smooth Ergodic Theory” Graduate Studies in Mathematics 148, AMS, 2013, 277 pp.

Gerhard Knieper

The main mechanism which causes complicated or chaotic behavior of a deterministic dynamical system is hyperbolicity. Hyperbolicity implies that trajectories have some sensitive dependence on initial conditions. Indications of this phenomenon were first discovered by Poincaré [16] in his study of the restricted 3-body problem. Not much later first papers on geodesic flows on surfaces of negative curvature appeared by Hadamard [7], Artin [2] and others. They discovered the complicated dynamics caused by the sign of curvature. In 1939, Hopf [8] showed in a fundamental paper that with respect to the Liouville measure the geodesic flow on compact surfaces of negative curvature is ergodic. The Liouville measure is the smooth measure induced by the canonical symplectic structure of the cotangent bundle. Since the geodesic flow can also be viewed as a Hamiltonian flow, with Hamiltonian function given by the Riemannian metric, the symplectic structure and therefore also the Liouville measure are flow invariant. Ergodicity is a very strong statistical property saying that all flow invariant L2-functions are constant almost everywhere. It particular, this implies that almost all orbits are dense in phase space. Crucial in Hopf’s proof are the stable and unstable foliations of the phase space whose stable resp. unstable leaves consist of manifolds which are exponentially contracted under the geodesic flow in forward resp. backward time. They are also geometrically described by the normal bundle of horospheres where the horospheres are the limits of geodesic spheres on the universal covering. In his proof, Hopf first observed that the time average of a continuous

G. Knieper (B) Ruhr-Universität Bochum, Bochum, Germany e-mail: [email protected] 166 G. Knieper function, which by Birkhoff’s ergodic theorem exists almost everywhere, is constant on the stable and unstable manifolds. Using this observation (nowadays called the Hopf argument) in combination with Fubini’s theorem, Hopf derived ergodicity of the geodesic flow, by showing that the time averages are constant almost everywhere. However, in order to apply Fubini’s theorem some regularity of the stable and unstable foliations is required. For surfaces he obtained a C1 regularity of the foliations which (for good reasons, see below) he was not able to extend to higher dimensions. The next main achievement came in 1962 with the seminal work of Anosov [1]. Not only did he resolve the difficulties encountered by Hopf caused by the lack of regularity, he also realized that all that was needed to prove ergodicity for flows with a smooth invariant measure is the uniform hyperbolicity guaranteed for geodesic flows by the negativity of the sectional curvature. A flow φt : X → X on a Riemannian manifold X is called uniformly hyperbolic (or Anosov) if there exists a decomposition

s u 0 TpX = E (p) ⊕ E (p) ⊕ E (p) of the tangent bundle TX into stable, unstable and neutral distribution E0(p) = d | t dt t=0φ (p) spanned by the infinitesimal generator of the flow. Furthermore, the distributions are invariant under the linearized system Dφt which contracts the stable resp. the unstable direction exponentially and uniformly in forward resp. backward time. More precisely, there are uniform constants c,λ such that − − − Dφt (p)ξ ≤ ce tλξ and Dφ t (p)η ≤ ce λt η (1) for all t ≥ 0, ξ ∈ Es and η ∈ Eu. Similarly a diffeomorphism F : X → X is called s u Anosov if there is a measurable splitting TpX = E (p) ⊕ E (p) into stable and unstable distributions such that linearized system DF n with discrete time n ∈ Z fulfills estimates analogous to the case of an Anosov flow. The Hadamard–Perron theorem allows to integrate the distribution and one obtains strong stable and unstable foliations. Given a smooth measure (i.e. a measure induced by some Riemannian metric) Anosov could show that the foliations are absolutely continuous. This means that for a set of positive measure A for almost all p ∈ A the intersection of the stable and unstable leaves through p with A have positive measure. Using this property together with Hopf’s argument Anosov obtained ergodicity for such systems. Anosov also found an obstruction for higher regularity of the stable and unstable distributions. Around 1975 Pesin [14] extended Anosov’s work by introducing a much more flexible notion for dynamical systems called nonuniform hyperbolicity. Unlike Anosov systems there are no obstructions that on a given compact manifold nonuniformly hyperbolic systems exist. In particular, the only compact surface admitting an Anosov diffeomorphism is the torus. In the discrete case the existence result has been obtained by Dolgopyat and Pesin [6] and was then extended to flows by Hu, Pesin and Talitskaya [9]. In the definition of nonuniformly hyperbolic systems one requires that the splitting s u 0 t TpX = E (p) ⊕ E (p) ⊕ E (p) of the tangent bundle TX holds for a φ -invariant measurable set Y ⊂ X. The constants c and λ in the estimates (1) above are replaced by positive measurable functions c,λ : Y → (0, ∞), where λ is φt -invariant and c is Luis Barreira and Yakov Pesin: “Introduction to Smooth Ergodic Theory” 167 allowed to increase along the orbits, however with an exponential rate smaller than λ. Also the angles between the stable and unstable distributions are allowed to decrease along the orbits with such a small exponential rate. Using the theory of Lyapunov exponents such hyperbolic sets are guaranteed for systems with hyperbolic invariant Borel probability measures defined as follows. For a given flow φt : X → X on a the Riemannian manifold X the Lyapunov exponent of a vector ξ ∈ TpX is given by

log Dφt (p)ξ χ(p,ξ):= lim sup . t→∞ t It is not difﬁcult to see that for any p ∈ X the function ξ → χ(p,ξ) can only attain ﬁnitely many values χ1(p) < ···<χs(p)(p), where s(p) is bounded by the dimension of X. By the fundamental muliplicative theorem of Oseledets for any invariant Borel probability measure μ almost all p ∈ X are Lyapunov regular which implies the existence of a splitting

TpX = E1(p) ⊕···⊕Es(p)(p) of the tangent space in linear subspaces defined by Ei(p) := ξ ∈ TpX | χ(p,ξ)= χi(p) and lim sup can be replaced by lim in the definition of χ(p,ξ). However, the linear spaces and the Lyapunov exponents depend only measurably on p. The space Ei(p) corresponding to vanishing Lyapunov exponent is called neutral and in the case of a flow it always contains the space spanned by the infinitesimal generator. The Lyapunov exponents and the regular points are flow invariant and the splitting is respected by the linearization. The measure μ is called hyperbolic in the flow case if for almost all Lyapunov regular points the neutral space is one dimensional (i.e. spanned by the infinitesimal generator). In the discrete case it is required that the neutral direction is trivial. If for a flow φt such a hyperbolic measure exists the system is nonuniformly hyperbolic, where the set Y is given by the Lyapunov regular points with nonvanishing Lyapunov exponents. Furthermore s u 0 d t E (p) = Ei(p), E (p) = χi(p) and E (p) = φ (p) dt t=0 χi (p)<0 χi (p)>0

As has been shown by Pesin there are subsets of almost full measure, called Pesin sets, where the stable and unstable bundles are continuous and the system is uniformly hyperbolic. Extending the work of Perron and Hadamard, Pesin was able to construct local stable and unstable manifolds. Furthermore, as in the uniform case they are absolutely continuous with respect to a smooth measure. Using the methods of Hopf, Pesin showed in case of a smooth hyperbolic measure that the system has only countably many ergodic components of positive measure. Under some extra condition on stable and unstable manifolds, which for example is fulfilled for some natural systems like the geodesic flow on surfaces of nonpositive curvature and genus 168 G. Knieper at least 2, Pesin showed that ergodic components of positive measure are open up to a set of measure zero. Assuming the existence of a dense orbit this implies ergodicity of the dynamical system restricted to the set with nonvanishing Lyapunov exponents. There are important relations between entropy and Lyapunov exponents. Given an invariant Borel measure μ the Kolmogorov–Sinai entropy of a diffeomorphism T : X → X (in case of a flow one takes the time 1 map) hμ(T ) can be estimated by the Ruelle inequality

hμ(T ) ≤ χi(p) dim Ei(p)dμ. X χi (p)>0

Pesin obtained that in case of a smooth invariant measure equality holds (Pesin formula). The book of Barreira and Pesin provides a systematic and thorough introduction into the ergodic theory of nonuniform hyperbolic systems not available in other text books. The book consists of two parts where the first part contains the core of the theory. The authors begin in the first chapter by introducing basic examples of uniformly and nonuniformly hyperbolic systems illustrating the fundamental phenomena for such systems. In Chap. 2 the general theory of Lyapunov exponents is developed. After discussing elementary but crucial properties of Lyapunov exponents they apply this to study asymptotic properties of linear differential equations. This is also related to the fundamental work of Perron [15]. Of importance is a necessary and sufficient condition for the existence of Oseledets decomposition which in the case of linear differential equations can be obtained from the Jordan normal form. Lyapunov exponents are also important in the study of infinite matrix products. The results obtained play a major role in later applications to continuous and discrete time dynamical systems. In Chap. 3 Lyapunov exponents are used to study the stability of dynamical systems under small perturbations. Crucial estimates for dynamical systems with nonzero Lyapunov exponents are derived in Chap. 4. In particular the notion of Lya- punov regular points is introduced. Furthermore, Hölder regularity of the stable and unstable distributions is established. In Chap. 5 the theory of cocyles over dynamical systems is derived. An important cocycle is induced by the linearization of a dynamical system. Cocycles play a central role in the proof of Oseledets famous multiplicative ergodic theorem [13] which is carried out in Chap. 6. An important goal is to control the behavior of trajectories near a hyperbolic orbit. This is done in Chap. 7 where the local stable and unstable manifolds are constructed using a generalization of the Hadamard–Perron theorem. In Chap. 8 the measure theoretic properties of the local stable and unstable manifolds relative to a smooth measure are discussed. In particular, for a nonuniformly hyperbolic system the absolute continuity of those manifolds is shown. The ergodic properties of a nonuniform hyperbolic system with a smooth invariant hyperbolic measure (almost all Lyapunov exponents are non zero) is treated in Chap. 9. In particular, it is shown that the number of ergodic components is finite and Luis Barreira and Yakov Pesin: “Introduction to Smooth Ergodic Theory” 169 under some further assumption even open up to a set of measure zero. Furthermore, a proof of the entropy formula stated above is given. In Chap. 10 ergodic properties of the geodesic flow on a compact surface M of genus at least two and nonpositive Gaussian curvature K are discussed. The Gauss– Bonnet theorem implies that some negative curvature has to be present. In particular, it is shown that the set 1 := v ∈ SM | lim sup K cv(s) ds < 0 t→∞ t has nonzero Lyapunov exponents and positive Liouville measure. As an application of the theory developed in this book the authors show ergodicity of the geodesic flow restricted to . It is a challenging open problem whether the complement of the set has zero measure. Note, that this was asserted in [14] but the argument contains a gap which up to now could not be filled. The generalization to higher dimensional nonpositively curved manifolds is not treated in this book. Here hyperbolicity is guaranteed by the rank 1 condition, i.e. the existence of one geodesic which does not admit an orthogonal parallel Jacobi field. Further dynamical and geometric properties of those flows are discussed in the book [3] and the survey [12]. In particular, important questions related to the growth rate and distribution of periodic orbits can be found in the research papers [10, 11]. Beyond nonpositive curvature, Donnay [5] and Burns and Gerber [4] constructed on S2 Riemannian metrics such that their geodesic flows are nonuniformly hyperbolic and ergodic. Even so by the Gauss–Bonnet theorem the Riemannian metric must have positive curvature, the hyperbolicity is caused by the negatively curved parts of the surface. Up to now it is not known whether nonuniformly hyperbolic geodesic flows of positive curvature can exist. Besides examples from Riemannian geometry there are plenty of other dynamical systems with hyperbolic behavior. In the class of billiard systems the famous Sinai billiard consisting of a torus with a disc removed provides an example. Another example is the Bunimovich stadium billiard consisting of two half-circles connected by a pair of parallel segments. The second part of the book contains material which supplements the theory of nonuniformly hyperbolic systems developed in the first part of the book. In particular, the cone technique is discussed which provides sufficient conditions for the hyperbolicity of symplectic dynamical systems. Furthermore, the more general concept of partial hyperbolicity is introduced. To summarize, the book is carefully written and essentially self-contained. It is highly recommendable as an introduction into the ergodic theory of hyperbolic dynamical systems. For the general theory the reader needs only some basic knowledge in real analysis, ordinary differential equations and measure theory. Besides that some familiarity with ergodic theory, differential geometry and Hamiltonian systems could be helpful depending on the applications which one has in mind.

References

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2. Artin, E.: Ein mechanisches System mit quasi-ergodischen Bahnen. Abh. Math. Semin. Univ. Hamb. 3, 170–175 (1924) 3. Ballmann, W.: Lectures on Spaces of Nonpositive Curvature. DMV Seminar, vol. 25. Birkhäuser- Verlag, Basel (1995) 4. Burns, K., Gerber, M.: Real analytic Bernoulli geodesic flows on S2. Ergod. Theory Dyn. Syst. 8, 27–45 (1989) 5. Donnay, V.: Geodesic flow on the two-sphere. Part I: Positive measure entropy. Ergod. Theory Dyn. Syst. 8, 531–553 (1989) 6. Dolgopyat, D., Pesin, Ya.: Every compact manifold carries a completely hyperbolic diffeomorphism. Ergod. Theory Dyn. Syst. 22, 409–432 (2002) 7. Hadamard, J.: Les surfaces à courbures opposé es et leurs lignes géodesiques. J. Math. Pures Appl. (5)4, 27–73 (1898) 8. Hopf, E.: Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipz., Math.-Phys. Kl. 91, 261–304 (1939) 9. Hu, H., Pesin, Ya., Talitskaya, A.: Every compact manifold carries a hyperbolic Bernoulli flow. In: Modern Dynamical Systems and Applications, pp. 347–358. Cambridge Univeristy Press, Cambridge (2004) 10. Knieper, G.: On the asymptotic geometry of nonpositively curved manifolds. Geom. Funct. Anal. 7, 755–782 (1997) 11. Knieper, G.: The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds. Ann. Math. 148, 291–314 (1998) 12. Knieper, G.: Hyperbolic dynamics and Riemannian geometry. In: Hasselblatt, B., Katok, A. (Hrsg.) Handbook of Dynamical Systems, vol. 1A, pp. 453–545. Elsevier, Amsterdam (2002) 13. Oseledets, V.: A muliplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197–221 (1968) 14. Pesin, Ya.: Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv. 32, 54– 114 (1977) 15. Perron, O.: Die Ordnungszahlen linearer Differentialgleichungssysteme. Math. Z. 31, 748–766 (1930) 16. Poincaré, H.: Sur le problème des trois corps et les equations de la dynamique. Acta Math. 13, 12–70 (1890)