Celebratio Mathematica

R. J. Duffin and R. Bott: “Impedance synthesis without use of transformers,” J. Appl. Phys. 20 : 8 (1949), pp. 816. MR 0037753 article

R. Bott: “Nondegenerate critical manifolds,” Ann. Math. (2) 60 : 2 (September 1954), pp. 248–261. MR 0064399 Zbl 0058.09101 article

R. Bott: “On torsion in Lie groups,” Proc. Natl. Acad. Sci. U.S.A. 40 : 7 (1954), pp. 586–588. MR 0062750 Zbl 0057.02201 article

R. Bott and H. Samelson: “The cohomology ring of \( G/T \),” Proc. Natl. Acad. Sci. U.S.A. 41 : 7 (July 1955), pp. 490–493. MR 0071773 Zbl 0064.25903 article

R. Bott: “On the iteration of closed geodesics and the Sturm intersection theory,” Commun. Pure Appl. Math. 9 : 2 (May 1956), pp. 171–206. MR 0090730 Zbl 0074.17202 article

R. Bott: “Homogeneous vector bundles,” Ann. Math. (2) 66 : 2 (September 1957), pp. 203–248. MR 0089473 Zbl 0094.35701 article

R. Bott: “The stable homotopy of the classical groups,” Ann. Math. (2) 70 : 2 (September 1959), pp. 313–337. See also Proc. Natl. Acad. Sci. U.S.A. 43:10 (1957). MR 0110104 Zbl 0129.15601 article

M. F. Atiyah, R. Bott, and A. Shapiro: “Clifford modules,” Topology 3 : Supplement 1 (July 1964), pp. 3–38. MR 0167985 Zbl 0146.19001 article

M. F. Atiyah	Related	Volume
Arnold Samuel Shapiro	Related

R. Bott and S. S. Chern: “Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections,” Acta Math. 114 : 1 (1965), pp. 71–112. A Russian translation was published in Matematika 14:2 (1970). MR 0185607 Zbl 0148.31906 article

R. Bott: “The index theorem for homogeneous differential operators,” pp. 167–186 in Differential and combinatorial topology: A symposium in honor of Marston Morse (Princeton, NJ, 1964). Edited by S. S. Cairns. Princeton Mathematical Series 27. Princeton University Press, 1965. MR 0182022 Zbl 0173.26001 incollection

Harold Calvin Marston Morse	Related
Stewart Scott Cairns	Related

M. F. Atiyah and R. Bott: “A Lefschetz fixed point formula for elliptic complexes, I,” Ann. Math. (2) 86 : 2 (1967), pp. 374–407. MR 0212836 Zbl 0161.43201 article

R. Bott: “A residue formula for holomorphic vector-fields,” J. Differ. Geom. 1 : 3–4 (1967), pp. 311–330. MR 0232405 Zbl 0179.28801 article

Let \( X \) be a holomorphic vector-field on the compact complex analytic manifold \( M \). In an earlier note, the behavior of \( X \) near its zeros was related to the characteristic numbers of the tangent bundle to \( M \), and the explicit form of this relation was computed in the most nondegenerate situation, that is, in the case of \( X \) vanished at isolated points to the first order. Our aim here is to extend this result in two directions. On the one hand we consider the characteristic numbers of more general bundles \( E \) over \( M \) on which \( X \) “acts”, and on the other hand we allow \( X \) to vanish along submanifolds of higher dimension but still only to the first order.

R. Bott: “Vector fields and characteristic numbers,” Michigan Math. J. 14 : 2 (1967), pp. 231–244. Dedicated to R. L. Wilder on his seventieth birthday. MR 0211416 Zbl 0145.43801 article

M. F. Atiyah and R. Bott: “A Lefschetz fixed point formula for elliptic complexes, II: Applications,” Ann. Math. (2) 88 : 3 (November 1968), pp. 451–491. MR 0232406 Zbl 0167.21703 article

R. Bott: “On a topological obstruction to integrability,” pp. 127–131 in Global analysis (Berkeley, CA, 1–26 July 1968). Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 16. American Mathematical Society (Providence, RI), 1970. MR 0266248 Zbl 0206.50501 incollection

Stephen Smale	Related
S. S. Chern	Related	Volume

R. Bott and G. Segal: “The cohomology of the vector fields on a manifold,” Topology 16 : 4 (1977), pp. 285–298. MR 0645730 Zbl 0387.57012 article

The smooth vector fields on a smooth manifold \( M \) form a Lie algebra \( \operatorname{Vect}(M) \) under the bracket. Gelfand and Fuchs [1968, 1969, 1970a, 1970b] have studied the Lie algebra cohomology of \( \operatorname{Vect}(M) \), which they define by means of a cochain algebra \( A(M) \), where \( A^k(M) \) is the vector space of continuous \( \mathbb{R} \)-multilinear maps \[ \operatorname{Vect}(M) \stackrel{\leftarrow k \rightarrow}{\times \cdots \times} \operatorname{Vect}(M) \to \mathbb{C} \] and the differential \( \operatorname{d}:A^k(M)\to A^{k+1}(M) \) is defined by the formula \[ \operatorname{d}\alpha(\xi_1,\dots,\xi_{k+1}) = \sum_{i < j} (-1)^{i+j-1} \alpha([\xi_i,\xi_j],\xi_1,\dots,\hat{\xi}_i,\dots,\hat{\xi}_j,\dots,\xi_{k+1}). \] ‘Continuous’ refers to the usual \( C^{\infty} \) topology on \( \operatorname{Vect}(M) \). (Actually Gelfand and Fuchs considered the cohomology with real coefficient, but we have found it convenient to change from \( \mathbb{R} \) to \( \mathbb{C} \).).

In this paper we shall prove that when \( M \) is either a compact manifold or the interior of a compact manifold with boundary the cohomology of \( \operatorname{Vect}(M) \) is the same as that of the space of continuous cross-sections of a certain natural fibre bundle \( E_M \) on \( M \) associated to its tangent bundle. The fibre of \( E_M \) is an open manifold \( F \) whose cohomology is that of \( \operatorname{Vect}(\mathbb{R}^n) \). The result was conjectured independently by Fuchs and the first author, and has also been proved by Haefliger [1976] and Trauber by different methods.

R. Bott: “Some recollections from 30 years ago,” pp. 33–39 in Constructive approaches to mathematical models: Proceedings of a conference in honor of R. J. Duffin (Pittsburgh, PA, 10–14 July 1978). Edited by C. V. Coffman and G. J. Fix. Academic Press (New York), 1979. MR 559485 Zbl 0467.58012 incollection

Charles Vernon Coffman	Related
Richard James Duffin	Related
George Joseph Fix	Related

R. Bott: “An equivariant setting of the Morse theory,” Enseign. Math., II. Sér. 26 : 3–4 (1980), pp. 271–278. Republished as Monographie de l’Enseignement Mathématique 30 (1982). MR 610527 Zbl 0481.58015 article

M. F. Atiyah and R. Bott: “The Yang–Mills equations over Riemann surfaces,” Philos. Trans. R. Soc. Lond., A 308 : 1505 (1983), pp. 523–615. MR 702806 Zbl 0509.14014 article

M. F. Atiyah and R. Bott: “The moment map and equivariant cohomology,” Topology 23 : 1 (1984), pp. 1–28. MR 721448 Zbl 0521.58025 article

R. Bott and C. Taubes: “On the rigidity theorems of Witten,” J. Am. Math. Soc. 2 : 1 (1989), pp. 137–186. MR 954493 Zbl 0667.57009 article

R. Bott: “The Dioszeger years (1923–1939),” pp. 11–26 in Collected papers, vol. 1: Topology and Lie groups. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1994. MR 1280034 incollection

R. Bott: Collected papers, vol. 1: Topology and Lie groups. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1994. MR 1280032 Zbl 0820.01026 book

R. Bott: “Autobiographical sketch,” pp. 3–9 in Collected papers, vol. 1: Topology and Lie groups. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1994. MR 1280033 incollection

R. Bott: Collected papers, vol. 2: Differential operators. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1994. MR 1290361 Zbl 0807.01033 book

R. Bott: Collected papers, vol. 3: Foliations. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1995. MR 1321886 book

R. Bott: Collected papers, vol. 4: Mathematics related to physics. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1995. MR 1321890 Zbl 0823.01011 book

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Raoul H. Bott

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