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: “The stable homotopy of the classical groups,” Proc. Natl. Acad. Sci. U.S.A. 43 : 10 (1957), pp. 933–935. See also Ann. Math. 70:2 (1959). MR 0102802 Zbl 0093.03401 article

R. Bott: “Quelques remarques sur les théorèmes de périodicité” [Some remarks on periodicity theorems], Bull. Soc. Math. Fr. 87 (1959), pp. 293–310. In French. Also published in Topologie algébrique et géométrie différentielle (1960). MR 0126281 Zbl 0124.38203 article

R. Bott and H. Samelson: “Applications of the theory of Morse to symmetric spaces,” Am. J. Math. 80 : 4 (October 1961), pp. 964–1029. Dedicated to Marston Morse on his 65th birthday. A correction was published in Am. J. Math. 83:1 (1961). MR 0105694 Zbl 0101.39702 article

Harold Calvin Marston Morse	Related
Hans Samelson	Related

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

M. F. Atiyah and R. Bott: “The index problem for manifolds with boundary,” pp. 175–186 in Differential Analysis: Papers presented at the international colloquium (Bombay, 7–14 January 1964). Tata Institute of Fundamental Research Studies in Mathematics 2. Oxford University Press (London), 1964. MR 0185606 Zbl 0163.34603 incollection

M. Atiyah and R. Bott: “On the periodicity theorem for complex vector bundles,” Acta Math. 112 : 1 (1964), pp. 229–247. MR 0178470 Zbl 0131.38201 article

The periodicity theorem for the infinite unitary group [Bott 1959] can be interpreted as a statement about complex vector bundles. As such it describes the relation between vector bundles over \( X \) and \( X\times S^2 \), where \( X \) is a compact space and \( S^2 \) is the 2-sphere. This relation is most succinctly expressed by the formula \[ K(X\times S^2) \simeq K(X)\otimes K(S^2), \] where \( K(X) \) is the Grothendieck group of complex vector bundles over \( X \). The general theory of these \( K \)-groups, as developed in [Atiyah and Hirzebruch 1961], has found many applications in topology and related fields. Since the periodicity theorem is the foundation stone of all this theory it seems desirable to have an elementary proof of it, and it is the purpose of this paper to present such a proof.

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 differential operators,” Bull. Am. Math. Soc. 72 : 2 (1966), pp. 245–250. MR 0190950 Zbl 0151.31801 article

The classical Lefschetz fixed point formula expresses, under suitable circumstances, the number of fixed points of a continuous map \( f:X\to X \) in terms of the transformation induced by \( f \) on the cohomology of \( X \). If \( X \) is not just a topological space but has some further structure, and if this structure is preserved by \( f \), one would expect to be able to refine the Lefschetz formula and to say more about the nature of the fixed points. The purpose of this note is to present such a refinement (Theorem 1) when \( X \) is a compact differentiable manifold endowed with an elliptic differential operator (or more generally an elliptic complex). Taking essentially the classical operators of complex and Riemannian geometry we obtain a number of important special cases (Theorem 2, 3). The first of these was conjectured to us by Shimura and was proved by Eichler for dimension one.

M. F. Atiyah, R. Bott, and L. Gårding: “Lacunas for hyperbolic differential operators with constant coefficients, I,” Acta Math. 124 : 1 (July 1970), pp. 109–189. A Russian translation was published in Usp. Mat. Nauk 26:2(158). MR 0470499 Zbl 0191.11203 article

Lars Gårding	Related
M. F. Atiyah	Related	Volume

M. Atiyah, R. Bott, and V. K. Patodi: “On the heat equation and the index theorem,” Invent. Math. 19 : 4 (1973), pp. 279–330. Dedicated to Sir William Hodge on his 70th birthday. Errata were published in Invent. Math. 28:3 (1975). A Russian translation was published in Matematika 17:6 (1973). MR 0650828 Zbl 0257.58008 article

M. F. Atiyah	Related	Volume
William V. D. Hodge	Related
Vijay Kumar Patodi	Related

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: “Morse theoretic aspects of Yang–Mills theory,” pp. 7–28 in Recent developments in gauge theories. Edited by G. ’t Hooft, C. Itzykson, A. Jaffe, H. Lehmann, P. Mitter, I. Singer, and R. Stora. NATO Advanced Study Institutes Series 59. Plenum Press (New York), 1980. incollection

Pronob K. Mitter	Related
Arthur Jaffe	Related
Raymond Stora	Related
Gerardus ’t Hooft	Related
Isadore M. Singer	Related	Volume
Claude Itzykson	Related
Harry Paul Lehmann	Related

R. Bott and L. W. Tu: Differential forms in algebraic topology. Graduate Texts in Mathematics 82. Springer (New York), 1982. A Russian translation was published as Differentsial’nye formy v algebraicheskoj topologii (1989). MR 658304 Zbl 0496.55001 book

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: 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

The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2003. Republished in 2009. MR 2136846 Zbl 1072.01021 book

F. Hirzebruch	Related	Volume
M. F. Atiyah	Related	Volume
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume

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Raoul H. Bott

Raoul Harry Bott: 24 September 1923 – 20 December 2005

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