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## Über dieses Buch

This book is the fifth and final volume of Raoul Bott’s Collected Papers. It collects all of Bott’s published articles since 1991 as well as some articles published earlier but missing in the earlier volumes. The volume also contains interviews with Raoul Bott, several of his previously unpublished speeches, commentaries by his collaborators such as Alberto Cattaneo and Jonathan Weitsman on their joint articles with Bott, Michael Atiyah’s obituary of Raoul Bott, Loring Tu’s authorized biography of Raoul Bott, and reminiscences of Raoul Bott by his friends, students, colleagues, and collaborators, among them Stephen Smale, David Mumford, Arthur Jaffe, Shing-Tung Yau, and Loring Tu. The mathematical articles, many inspired by physics, encompass stable vector bundles, knot and manifold invariants, equivariant cohomology, and loop spaces. The nonmathematical contributions give a sense of Bott’s approach to mathematics, style, personality, zest for life, and humanity. In one of the articles, from the vantage point of his later years, Raoul Bott gives a tour-de-force historical account of one of his greatest achievements, the Bott periodicity theorem. A large number of the articles originally appeared in hard-to-find conference proceedings or journals. This volume makes them all easily accessible.
It also features a collection of photographs giving a panoramic view of Raoul Bott's life and his interaction with other mathematicians.

## Inhaltsverzeichnis

Loring W. Tu

### Curriculum Vitae

Raoul Bott, Loring W. Tu

### Obituary: Raoul Harry Bott, FRS, 1923–2005

Raoul Bott was born in Budapest on 24 September 1923 but lived, until the age of 16 years, in Slovakia. He was a typical child of the Austro-Hungarian world, speaking and educated in a variety of languages: Hungarian, Slovak, German and English (learnt from his English governesses). His mother was Hungarian and Jewish, whereas his father was Austrian and Catholic. Despite the fact that his parents’ marriage broke up shortly after he was born, and that he saw very little of his father, his mother brought him up as a Catholic, and (though lapsing as a teenager) he remained one throughout his life.

Sri Michael Atiyah

Loring W. Tu

Loring W. Tu

### Memories of Raoul Bott

In fall ’69 I arrived in the Boston area driving across the country from Berkeley to take up a fellowship at MIT. A big attraction was to learn more about differential geometry mathematics from Raoul Bott. Right away Raoul tried to help me find an apartment in Cambridge.

Dennis Sullivan

### Memories of Raoul Bott

In 1973–74 I had the good fortune to be introduced to topology by Raoul Bott. Let me try to convey something of what that was like and how the experience has stayed with me over the last forty years. Twice a week we would take our seats in the classroom, and there he would be: standing tall in front of us with a piece of chalk in his hand and a bit of a twinkle in his eye. He had a commanding presence, but never a formidable one. Above all it was a presence: he was so much with us! His love of the subject was unmistakable. Not that he talked about the power and the glory and the beauty of mathematics: he simply talked about mathematics, and it was beautiful and glorious and powerful.

Thomas G. Goodwillie

### Stable Bundles (Commentary on [95])

The article “Stable Bundles Revisited” is a review article describing the fundamental work of Atiyah and Bott in [1], where the authors prove formulas for the Poincaré polynomial of the moduli space M(n, d) of stable bundles of rank n and degree d over a Riemann surface. Atiyah and Bott are motivated by Morse theory of the normsquare of the moment map for the action of gauge group on the space of all connections on the surface (an infinite-dimensional vector space). Atiyah and Bott made the fundamental observation that the action of the gauge group on the space of all connections is Hamiltonian and the moment map is the curvature.

Lisa Jeffrey

### On E. Verlinde’s Formula (Commentary on [96])

The article “On E. Verlinde’s Formula in the Context of Stable Bundles” is a review article describing the Verlinde formula (a formula discovered by the physicist E. Verlinde [10]), which is a formula for the dimension of the space of holomorphic sections of a line bundle over the moduli space M(n, d) of gauge equivalence classes of stable bundles of coprime rank n and degree d over a Riemann surface Σ g of genus g ≥ 2.All holomorphic line bundles over this space are obtained as the k-th power of a generating line bundle L.

Lisa Jeffrey

### Memories of Raoul Bott (Commentary on [96])

Raoul Bott was a wonderful lecturer, with a talent to get to the essence of the matter, masterfully evading most technical complications. For the uninitiated, this often had the effect of creating the illusion of understanding everything. This understanding then painfully disappeared the next morning, replaced by a slew of questions, and this is where the actual learning from the talk began.

András Szenes

### On Raoul Bott’s “On Invariants of Manifold” (Commentary on [106], [107])

I’m not sure how to introduce a review paper [B]. So rather than commenting on the paper as whole, I will concentrate on my subjective view of just one paragraph—a paragraph which I think I influenced and which ended up influencing me very deeply. A paragraph I am sure Raoul was uncomfortable writing, for at the time he was uncomfortable with his understanding of the underlying mathematics as I have explained it to him [BN]—uncomfortable enough to later rewrite (with Taubes) this bit of mathematics in his own language [BT], making my own work completely obsolete.

Dror Bar-Natan

### Configuration Space Integrals: Bridging Physics, Geometry, and Topology of Knots and Links (Commentary on [106], [108], [109])

Early 1990s witnessed an emergence of new techniques and points of view in the study of spaces of knots and spaces of embeddings more generally. One of the most exciting developments was the introduction of finite type or Vassiliev knot invariants [Vas90]. To explain, any knot invariant V can be extended to singular knots with n transverse double points via the repeated use of the Vassiliev skein relation

Ismar Volić

### Integral Invariants of 3-Manifolds (Commentary on [111], [114])

These papers arose from my collaboration with Raoul during my stay at Harvard. I was there as a postdoc in physics, in the group of Arthur Jaffe. It was the critical time when I had to decide what to do with my life, including whether I should complete my move to mathematics. At the time I was working on perturbative topological field theories and one day, while I was alone in my office, I was visited by a tall guy who said he was interested in what I was doing and wished to start a collaboration to apply my results.

Alberto S. Cattaneo

### Equivariant Characteristic Classes (Commentary on [116])

I was trained as an algebraic geometer under Phillip A. Griffiths, but I have always had an abiding interest in topology, especially Raoul Bott’s kind of topology. In 1995 Raoul Bott gave a series of lectures at Brown University on equivariant cohomology. I was very much captivated by his presentation of the subject matter.

Loring W. Tu

Loring W. Tu

### Commentary on “Surjectivity for Hamiltonian Loop Group Spaces” [120]

Two of Raoul Bott’s major works (B; AB) study Morse theory in two apparently unrelated settings. In (BTW), we show that these results fit into a general theorem about Hamiltonian actions of loop groups.

Jonathan Weitsman

### Woods Hole (Commentary on [124])

This short article [124], written with Raoul, brings back happy memories of the Woods Hole summer conference of 1964. How we envied the oceanographers! I remember the lively atmosphere, full of active young mathematicians—alas this was nearly 50 years ago.

Michael Atiyah

Loring W. Tu

### [95] Stable Bundles Revisited

The topological classification of complex vector bundles over a Riemann surface is of course very simple: they are classified by one integer c1(E), corresponding to the first Chern class of E. On the other hand, a classification in the complex-analytic—or algebro-geometric—category leads to “continuous moduli” and subtle phenomena which have links with number theory, gauge theory, and conformal field theory. I will try to report briefly on some of these developments here.

Loring W. Tu, Raoul Bott

### [96] On E. Verlinde’s Formula in the Context of Stable Bundles

E. Verlinde’s formula for the dimension of the nonabelian θ-functions is discussed from an algebraic geometry point of view and related to certain quotient rings of the representative ring of sums.

Raoul Bott

### [98] Topological Aspects of Loop Groups

The purpose of these lectures is to give an introduction to the topological aspect s of the loop space ΩG when G is a compact Lie group. We will give a direct method of computing the cohomology of ΩG from very geometric and group theoretic data, usually referred to as the diagram. The main tool in our calculations is a version of Morse theory adapted to the study of loop spaces.

László Fehér, András Stipsicz, János Szenthe

Raoul Bott

Loring W. Tu

Loring W. Tu

### [102] Life on the Slippery Lane—Between Mathematics and Physics

Last week when many of us were listening to Andy’s official “last” lecture, I could not help recalling the first time I heard him. Unbelievably it was thirty-five years ago—in 1958 at Michigan. I remembered his energy, his beautiful writing, and above all that certain something that always alerts us to the fact that we are in the presence of a master.

Raoul Bott

### [103] Reflections on the Theme of the Poster

It is a pleasure to be allowed to speak at this celebration of Jack Milnor’s “coming of age”. For aren’t these occasions reserved for the younger generation, the students of the honoree? But of course under the rubric of Jack’s students we are all eminently qualified, regardless of age.

Raoul Bott, Loring W. Tu

### [104] Luncheon Talk and Nomination for Stephen Smale

Thank you, Moe. It is a great pleasure to be here, but I must say that I feel a bit like Karen who just remarked that she comes from a different world. I come from the beach! In fact, I’m hardly dry. On my beach, you don't usually wear clothes, but as you see, I put some on for this great occasion.

R. Bott, Loring W. Tu

### [105] The Work of Robert D. MacPherson

Robert D. MacPherson has received the National Academy of Sciences Award in Mathematics, a prizeof \$5000 established by the AMS to commemorate the Society’s centennial in 1988. Professor MacPherson wasrecognized for “his role in the introduction and application of radically new approaches to the topologyof singular spaces including characteristic c lasses, intersection homology, perverse sheaves, and stratified Morse theory. “ The award was one of thirteen presented at the Academy’s I 29th meeting on April 27,1992.

Loring W. Tu

### [106] On the Self-Linking of Knots

This note describes a subcomplex F of the de Rham complex of parametrized knot space, which is combinatorial over a number of universal “Anomaly Integrals.” The self-linking integrals of Guadaguini, Martellini, and Mintchev [“Perturbative aspects of Chem–Simons field theory,” Phys . Lett. B 227, 111 (1989)] and BarNatan [“Perturbative aspects of the Chem–Simons topological quantum field theory,” Ph.D. thesis, Princeton University, 1991; also “On the VassiUev Knot Invariants” (to appear in Topology)] are seen to represent the first nontrivial element in H0 (F)—occurring at level 4, and are anomaly free. However, already at the next level an anomalous term is possible.

Raoul Bott, Clifford Taubes

### [107] On Invariants of Manifolds

Raoul Bott, Loring W. Tu

### [108] Configuration Spaces and Imbedding Invariants

Loring W. Tu, Raoul Bott

Raoul Bott

### [110] Critical Point Theory in Mathematics and in Mathematical Physics

At last year’s Gökova Conference I reported on a “topological approach” to the “new” knot invariants, which ClifF Taubes and I had worked out along lines initiated by Axelrod and Singer [AS] and Kontsevich [K] in their work on 3-manifold invariants.

Raoul Bott

### [111] Integral Invariants of 3-Manifolds

This note describes an invariant of rational homology 3-spheres in terms of configuration space integrals, which in some sense lies between the invariants of Axelrod and Singer [2] and those of Kontsevich [9].

Raoul Bott, Alberto S. Cattaneo

### [112] Lars Valerian Ahlfors (1907–1996)

Lars Valerian Ahlfors was arguably the preeminent complex function theorist of the twentieth century. With a career spanning more than sixty years, Ahlfors made decisive contributions to areas ranging from meromorphic curves to value distribution theory, Riemann surfaces, conformal geometry, extremal length, quasiconformal mappings, and Kleinian groups ([7] serves as a map of Ahlfors’s contributions to the subject). Ahlfors was both role model and mentor to his graduate students and to the many mathematicians around the world who learned from his example. He is remembered warmly, both as a mathematician and as a man.

Raoul Bott, Clifford Earle, Dennis Hejhal, James Jenkins, Troels Jorgensen, Steven G. Krantz, Albert Marden, Robert Osserman

### [113] An Introduction to Equivariant Cohomology

It gives me great pleasure to start off this “anniversary” meeting, even if by default.The 1970 summer school here at Les Houches was a very memorable event for me and my whole family and it is quite wonderful and magical to see, among the many young and eager faces, so many friends of old. In the nearly 30 years since 1970, I am very pleased to report that the dialogue between physiCs and mathematics has increased dramatically, and, on the mathematical side at least, this interaction has been highly productive. The Yang-Mills theory spawned the Donaldson invariants of four-manifolds, later to be augmented by the Seiberg- Witten invariants. Current algebras engendered a new and beautiful representation theory of Loop groups and Kac- Moody algebras. Knot theory has been reinvigorated by the “ChernSimons Theory,” while 19th century questions in algebraic geometry have been solved by methods initiated by “String Theory.” And, on a personal note, I might add that nowadays it is usually the Physics graduate students who are the stars in my geometry courses.

R. Bott

### [114] Integral Invariants of 3-Manifolds, II

This note is a sequel to our earlier paper of the same title [4] and describes invariants of rational homology 3-spheres associated to acyclic orthogonal local systems. Our work is in the spirit of the Axelrod–Singer papers [1], generalizes some of their results, and furnishes a new setting for the purely topological implications of their work.

Raoul Bott, Alberto S. Cattaneo, Loring W. Tu

### [115] A Remark on Integral Geometry

Raoul Bott, Cliff Taubes

### [116] Equivariant Characteristic Classes in the Cartan Model

There is also a differential geometric definition of equivariant characteristic classes in terms of the curvature of a connection on P (3)(4). However, there does not seem to be an explanation or justification in the literature bridging the two approaches. The modest purpose of this note is to show the compatibility of the usual differential geometric formulation of equivariant characteristic classes with the topological one. We have also tried to be as self-contained as possible, which partly explains the length of this article.

Raoul Bott, Loring W. Tu

### [117] Interview with Raoul Bott

This is the edited text of two interviews with Raoul Bott, conducted by Allyn Jackson in October 2000.

Loring W. Tu

### [118] Response to Shimura’s Letter

I recently noticed the following passage in “Interview with Raoul Bott”, Notices vol. 48, No. 4 (April 2001), p. 379.

Goro Shimura, Raoul Bott

### [119] Introduction to Woods Hole Mathematics: Perspectives in Mathematics and Physics

Woods Hole has played such a vital role in both my mathematical and personal life that it is a great pleasure to see the mathematical tradition of the 1964 meeting resurrected forty years later and, as this volume shows, resurrected with new vigor and one hopes on a regular basis. I therefore consider it a signal honor to have been asked to introduce this volume with a few reminiscences of that meeting forty years ago.

Nils Tongring, R C Penner

### [120] Surjectivity for Hamiltonian Loop Group Spaces

Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X, ω) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X, ω), and assume that the moment map μ:Χ→Lg∗ $$\mu \,:\,\,{\rm X}\,\, \to L{g^ * }$$ is proper. We consider the function |μ|2:X→ℝ $${\left| \mu \right|^2}:\,X\, \to \,\mathbb{R}$$ , and use a version of Morse theory to showthat the inclusion map j:μ−1(0)→X $$j\,:\,{\mu ^{ - 1}}\left( 0 \right)\, \to \,X$$ induces a surjection j∗:HG∗(X)→HG∗(μ−1(0)) $$j{\,^ * }:\,\,H_G^ * \left( X \right)\, \to \,H_G^ * \left( {{\mu ^{ - 1}}\left( 0 \right)} \right)$$ , in analogywithKirwan’s surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian G-spaces.

Raoul Bott, Susan Tolman, Jonathan Weitsman

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### [128] On Mathematics, Commencement Lecture at the University of Costa Rica

Loring W. Tu
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