Celebratio Mathematica

R. Bott: Electrical network theory. Ph.D. thesis, Carnegie Mellon University, 1949. Advised by R. J. Duffin. MR 2937907 phdthesis

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: “On the third symmetric potency of \( S_1 \),” Fundam. Math. 39 : 1 (1952), pp. 264–268. From a letter to K. Borsuk. MR 0054954 Zbl 0050.17801 article

R. Bott: “Two new combinatorial invariants for polyhedra,” Port. Math. 11 : 1 (1952), pp. 35–40. MR 0048820 Zbl 0047.42003 article

In this note we introduce two new combinatorial invariants for polyhedral complexes. The idea behind their construction is the following one: By removing the \( n \)-cells (their interiors) from a given finite \( n \)-complex \( \pi \), we generate a family of subcomplexes \( \pi_s \). The invariants we introduce consist of certain expressions summed over the whole family \( |\pi_s| \). The form of the expressions to be averaged in this way is suggested by studying how the family \( |\pi_s| \) changes when we go over to an elementary subdivision of \( \pi \). (An elementary subdivision just splits an \( n \)-cell in two).

The new invariants are therefore definitely « cooked up », and not much geometrical good can be expected of them. They are presened here, then, mainly as a curiosity.

The most interesting feature of the \( R \) and \( S \) polynomials (as we call our invariants) of a complex \( \pi \), is that they are not invariants of the homotopy-type of \( \pi \). However it is this very property which makes it unlikely that the topological invariance of the polynomials \( R \) and \( S \) will be decided before the « Hauptvermutung » is.

R. Bott and H. Samelson: “On the Pontryagin product in spaces of paths,” Comment. Math. Helv. 27 (1953), pp. 320–337. MR 0060233 Zbl 0052.19301 article

For a topological (arcwise connected) space \( X \), let \( E \) be the function space consisting of the paths in \( X \) which start at a certain point \( x \), and let \( \Omega \) be the subspace of \( E \) consisting of the closed paths or loops; these spaces have been studied in particular by M. Morse [1934] and J.-P. Serre [1951]; \( E \) is a fiber space over \( X \). Now \( \Omega \) admits a natural multiplication: two loops in succession make a new loop (actually there is a more general operation between \( E \) and \( \Omega \)). This multiplication gives rise to a multiplication of the elements of the homology group of \( \Omega \); we call this the Pontryagin-multiplication, since an entirely similar concept for group spaces was introduced by Pontryagin [1939] (cf. also [Samelson 1941]). In part II we study the relation between Leray’s spectral sequence of \( E \) (in Serre’s formulation [1951]) and the Pontryagin product. A closely related situation, involving compact Lie groups and Leray’s cohomology theory, has been considered by Leray [1949a, 1949b] and A. Borel [1953]; our results are analogous to theirs. In part III we determine, as application, the Pontryagin ring of a space which is union of several spheres with a point in common. We have to consider the homology of Cartesian product spaces. This makes necessary a modification of Serre’s cubic homology theory; we present this in part I. Generally speaking we follow Serre’s definitions, notation and conventions, with some minor deviations; we assume familiarity with his paper.

R. Bott and R. J. Duffin: “On the algebra of networks,” Trans. Am. Math. Soc. 74 : 1 (1953), pp. 99–109. MR 0056573 Zbl 0050.25104 article

R. Bott: “On symmetric products and the Steenrod squares,” Ann. Math. (2) 57 : 3 (May 1953), pp. 579–590. MR 0056294 Zbl 0053.30104 article

In [1952] N. E. Steenrod defines a set of invariant cohomology operations. These “reduced powers” include as a special case the squares which Steenrod introduced earlier [1947].

As the definition of the reduced power depends essentially on a transformation group in the product space \( K\times K\times \cdots \times K \) (\( p \) factors) of a complex \( K \) it seemed likely that at least in certain cases a close relation between his operations and the Smith–Richardson theory of symmetric products would exist. (See [Richarson 1935], [Richardson and Smith 1938] and [Smith 1933].)

In this paper we will discuss the case \( p = 2 \), and obtain a defintion of the Steenrod squares directly from the exact sequences which are associated with the symmetric product of a space.

Our main result can be stated as follows: given a complex \( X \), set \( K = X\times X \) and let \( k \) be the space obtained from \( K \) by identifying points which correspond to each other under the involution which exchanges the factors of \( K \). Let \( l \) be the image in \( k \) of the diagonal of \( K \).

R. Bott: “Symmetric solutions to majority games,” pp. 319–323 in Contributions to the theory of games, vol. 2. Edited by H. W. Kuhn and A. W. Tucker. Annals of Mathematics Studies 28. Princeton University Press, 1953. MR 0053475 Zbl 0050.14405 incollection

Harold William Kuhn	Related
Albert William Tucker	Related

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

R. Bott and J. P. Mayberry: “Matrices and trees,” pp. 391–400 in Economic activity analysis. Edited by O. Morgenstern. John Wiley (New York), 1954. MR 0067067 Zbl 0057.12402 incollection

Oskar Morgenstern	Related
John Patterson Mayberry	Related

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: “On manifolds all of whose geodesics are closed,” Ann. Math. (2) 60 : 3 (November 1954), pp. 375–382. MR 0073993 Zbl 0058.15604 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: “An application of the Morse theory to the topology of Lie-groups,” Bull. Soc. Math. Fr. 84 (1956), pp. 251–281. MR 0087035 Zbl 0073.40001 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: “Homogeneous vector bundles,” Ann. Math. (2) 66 : 2 (September 1957), pp. 203–248. MR 0089473 Zbl 0094.35701 article

R. Bott and H. Samelson: “Applications of Morse theory to symmetric spaces,” pp. 282–284 in Symposium internacional de topología algebraica [International symposium on algebraic topology] (Mexico City, 1956). Universidad Nacional Autónoma de México and UNESCO (Mexico City and Paris), 1958. Zbl 0101.39701 incollection

R. Bott, T. H. Hildebrandt, R. K. Ritt, E. H. Rothe, and H. Samelson: “In memoriam Sumner B. Myers 1910–1955,” Mich. Math. J. 5 : 1 (1958), pp. 1–4. MR 0096571 Zbl 0082.24316 article

Hans Samelson	Related
Sumner Byron Myers	Related
Robert King Ritt	Related
Erich Hans Rothe	Related
Theophil Henry Hildebrandt	Related

R. Bott: “The space of loops on a Lie group,” Mich. Math. J. 5 : 1 (1958), pp. 35–61. MR 0102803 Zbl 0096.17701 article

R. Bott and J. W. Milnor: “On the parallelizability of the spheres,” Bull. Am. Math. Soc. 64 : 3 (1958), pp. 87–89. MR 0102804 Zbl 0082.16602 article

R. Bott: “Induced representations,” pp. 88–91 in Seminars on analytic functions (Princeton, NJ, 2–14 September 1957), vol. 2. Institute for Advanced Study (Princeton, NJ), 1958. Zbl 0116.40303 incollection

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: “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

R. Bott: “On a theorem of Lefschetz,” Mich. Math. J. 6 : 3 (1959), pp. 211–216. MR 0215323 Zbl 0113.36502 article

This note concerns the Lefschetz hyperplane theorem in both homology and homotopy. (See Theorem I and its corollary, for the statement of the result.) We shall deduce our refinement of this oft-proved theorem (see [Bott 1959; Fáry 1957; Lefschetz 1924; Wallace 1958]) as an immediate consequence of what in [1959] I called the main theorem of the nondegenerate Morse theory.

Morse and Lefschetz lived within a few steps of one another for over twenty years. It is therefore amusing that the idea of applying the former’s theory in this connection is quite recent. I first saw this approach taken in a lecture by R. Thom in 1957, and this note is no more than a technical elaboration of his idea. My main observation is that the notion of a nondegenerate critical manifold, when properly applied, eliminates all the troubles with infinity which Thom encountered in his original account.

The proper dual of the Lefschetz theorem states that the homology of a Stein manifold vanishes above its middle dimension. In a forthcoming paper [1959], Andreotti and Frankel use the Morse theory in its most elementary form to prove this dual statement. Poincaré duality then completes their proof of the classical version of the Lefschetz theorem. This approach is in a sense the simplest. But it does not yield the homotopy statement.

R. Bott: “A note on the Samelson product in the classical groups,” Comment. Math. Helv. 34 (1960), pp. 249–256. MR 0123330 Zbl 0094.01503 article

R. Bott: “Quelques remarques sur les théorèmes de périodicité” [Some remarks on periodicity theorems], pp. 293–310 in Topologie algébrique et géométrie différentielle [Algebraic topology and differential geometry] (1–6 June 1959, Lille). Colloques Internationaux du CNRS 89. CNRS (Paris), 1960. In French. Also published in Bull. Soc. Math. Fr. 87 (1959). Zbl 0108.35903 incollection

R. Bott: “An application of the Morse theory to the topology of Lie groups,” pp. 423–426 in Proceedings of the International Congress of Mathematicians 1958 (Edinburgh, 14–21 August 1958). Edited by J. A. Todd. Cambridge University Press, 1960. MR 0117746 Zbl 0116.40204 incollection

R. Bott: “Vector fields on spheres and allied problems,” Enseignement Math. (2) 7 (1961), pp. 125–138. MR 0139179 Zbl 0105.17201 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

R. Bott: “A report on the unitary group,” pp. 1–6 in Differential geometry: Proceedings of the third symposium in pure mathematics of the American Mathematical Society (Tucson, AZ, 18–19 February 1960). Edited by C. B. Allendoerfer. Proceedings of Symposia in Pure Mathematics 3. American Mathematical Society (Providence, RI), 1961. MR 0132553 Zbl 0117.40301 incollection

R. Bott and H. Samelson: “Correction to ‘Applications of the theory of Morse to symmetric spaces’,” Am. J. Math. 83 : 1 (January 1961), pp. 207–208. Correction to an article published in Am. J. Math. 80:4 (1961). MR 0170351 article

M. Berger and R. Bott: “Sur les variétés à courbure strictement positive” [On varieties of strictly positive curvature], Topology 1 : 4 (October–December 1962), pp. 301–311. In French. MR 0146776 Zbl 0112.13604 article

R. Bott: “A note on the \( KO \)-theory of sphere-bundles,” Bull. Am. Math. Soc. 68 : 4 (1962), pp. 395–400. MR 0153019 Zbl 0115.17301 article

R. Bott: Lectures on \( K(X) \). Mathematics Lecture Note Series. Harvard University (Cambridge, MA), 1963. With an appendix on a spectral sequence for \( K \)-theory by Emery Thomas, and an appendix on the periodicity theorem for vector bundles by M. Atiyah and R. Bott. Republished in 1969. book

M. F. Atiyah	Related	Volume
P. Emery Thomas	Related	Volume

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.

M. F. Atiyah and R. Bott: Notes on the Lefschetz fixed point theorem for elliptic complexes. Harvard University (Cambridge, MA), 1964. Zbl 0161.43101 book

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.

R. Bott: “A fixed point theorem for elliptic complexes,” pp. 6–12 in Proceedings of the United States-Japan seminar in differential geometry (Kyoto, 14–19 June 1965). Nippon Hyoronsha (Tokyo), 1966. MR 0215318 Zbl 0149.41202 incollection

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 and J. Mather: “Topics in topology and differential geometry,” pp. 460–515 in Battelle rencontres: 1967 lectures in mathematics and physics (Seattle, WA, 16 July–31 August 1967). Edited by C. M. DeWitt and J. A. Wheeler. W. A. Benjamin (New York), 1968. Zbl 0177.26001 incollection

John N. Mather	Related
Cécile M. DeWitt-Morette	Related
John Archibald Wheeler	Related

R. Bott: Lectures on \( K(X) \). Mathematics Lecture Note Series. W. A. Benjamin (New York and Amsterdam), 1969. Republication of 1963 original. MR 0258020 Zbl 0194.23904 book

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

R. Bott and S. S. Chern: “Some formulas related to complex transgression,” pp. 48–57 in Essays on topology and related topics: Mémoires dédiés à Georges de Rham [Essays on topology and related topics: Memoirs dedicated to Georges de Rham]. Edited by A. Haefliger and R. Narasimhan. Springer (New York), 1970. MR 0264715 Zbl 0203.54202 incollection

Let \( X \) be a complex manifold of complex dimension \( n \) and \( \pi:E\to X \) a holomorphic vector bundle whose fiber dimension is also \( n \). On \( E \) we introduce a positive definite hermitian norm \( N \) and denote by \[ B^*(E) = \{e\in E\mid 0 < N(e)\} \] the subset of non-zero vectors of \( E \). Let \( c_n(E) \) be the Chern form of the hermitian structure (to be described again below). In [1965] we proved the theorem: There exists a real-valued form \( \rho \) of type \( (n-1, \) \( n-1) \) on \( B^*(E) \) such that \begin{equation*}\tag{1} \pi^* c_n(E) = \frac{dd^c}{4\pi}\rho. \end{equation*} In this paper we wish to give a proof of (1) based on an explicit construction. We will follow and extend a formalism developed in the real case by H. Flanders [1953]. It is possible that this formalism will be useful for later purposes.

S. S. Chern	Related	Volume
Georges de Rham	Related
Raghavan Narasimhan	Related
André Haefliger	Related

M. Atiyah and S. S. Chern: “Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections,” Matematika, Moskva 14 : 2 (1970), pp. 117–154. Russian translation of an article in Acta Math. 114:1 (1965). Zbl 0208.35101 article

M. F. Atiyah	Related	Volume
S. S. Chern	Related	Volume

P. F. Baum and R. Bott: “On the zeroes of meromorphic vector-fields,” pp. 29–47 in Essays on topology and related topics: Mémoires dédiés à Georges de Rham [Essays on topology and related topics: Memoirs dedicated to Georges de Rham] (Geneva, 26–28 March 1969). Edited by A. Haefliger and R. Narasimhan. Springer (Berlin), 1970. MR 261635 Zbl 0193.52201 incollection

Let \( M \) be a compact complex analytic manifold and let \( x \) be a holomorphic vector-field on \( M \). In an earlier paper by one of us (see [Bott 1967]) it was shown that the behavior of \( x \) near its zeroes determined all the Chern numbers of \( M \) and the nature of this determination was explicitly given where \( x \) had only nondegenerate zeroes. The primary purpose of this note is to extend this result to meromorphic fields, or equivalently to sections \( s \) of \( T\otimes L \) where \( T \) is the holomorphic tangent bundle to \( M \) and \( L \) is a holomorphic line bundle. We will also drop the non-degeneracy assumption of the zeroes of \( s \), but we treat only the case where \( s \) vanishes at isolated points \( \{p\} \).

Georges de Rham	Related
Paul Baum	Related	Volume
Raghavan Narasimhan	Related
André Haefliger	Related

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: “The periodicity theorem for the classical groups and some of its applications,” Adv. Math. 4 : 3 (June 1970), pp. 353–411. MR 0259904 Zbl 0231.55010 article

M. F. At’ja, R. Bott, and L. Gording: “Lacunas for hyperbolic differential operators with constant coefficients, I,” Usp. Mat. Nauk 26 : 2(158) (1971), pp. 25–100. Russian translation of an article in Acta Math. 124:1 (1970). MR 0606062 Zbl 0208.13201 article

Lars Gårding	Related
M. F. Atiyah	Related	Volume

R. Bott: “On topological obstructions to integrability,” pp. 27–36 in Actes du Congrès International des Mathématiciens [Proceedings of the International Congress of Mathematicians] (Nice, 1970), vol. 1. Gauthier-Villars (Paris), 1971. MR 0425983 Zbl 0225.57010 incollection

In this lecture I would like to describe the state of the art in the problem of “foliating” a manifold or, as I prefer to view it, the problem of constructing integrable fields on a manifold. The subject has seen some interesting developments in the past two years and is also contemporary in the sense that, as you will see, it leads to “huge spaces”. By a huge space I mean here simply one whose homotopy groups are not finitely generated in every dimension. In the past we — and I think quite rightly — have shied away from such objects, but recently they have cropped up in various contexts: notably in the index theory associated to Von Neumann algebras of type II, and also in the localization of spaces at a given prime, and I am confident that in the future these “huge” spaces will enter into many of the analysis inspired problems in topology.

R. Bott: “Lectures on characteristic classes and foliations,” pp. 1–94 in Lectures on algebraic and differential topology (Mexico City, July 1971). Edited by S. Gitler. Lecture Notes in Mathematics 279. Springer (Berlin), 1972. Second Latin American School in Mathematics. Notes by Lawrence Conlon, with two appendices by J. Stasheff. MR 0362335 Zbl 0241.57010 incollection

Samuel C. Gitler	Related
James D. Stasheff	Related
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R. Bott: “On the Lefschetz formula and exotic characteristic classes,” pp. 95–105 in Convegni del Maggio del 1971 e del Gennaio del 1972 [Conferences of May 1971 and January 1972] (Rome, 24–28 May 1971 and 10–13 January 1972). Symposia Mathematica 10. Academic Press (London), 1972. MR 0362336 Zbl 0254.57013 incollection

R. Bott and J. Heitsch: “A remark on the integral cohomology of \( B\Gamma_q \),” Topology 11 : 2 (May 1972), pp. 141–146. MR 0293661 Zbl 0236.55015 article

R. Bott and A. Haefliger: “On characteristic classes of \( \Gamma \)-foliations,” Bull. Am. Math. Soc. 78 : 6 (November 1972), pp. 1039–1044. MR 307250 Zbl 0262.57010 article

P. Baum and R. Bott: “Singularities of holomorphic foliations,” J. Diff. Geom. 7 : 3–4 (1972), pp. 279–342. To S. S. Chern and D. C. Spencer on their 60th birthdays. MR 377923 Zbl 0268.57011 article

Donald Clayton Spencer	Related
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M. Atiyah, R. Bott, and V. K. Patodi: “On the heat equation and the index theorem,” Matematika, Moskva 17 : 6 (1973), pp. 3–48. Russian translation of an article in Invent. Math. 19:4 (1973). Zbl 0364.58016 article

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

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

M. F. Atiyah, R. Bott, and L. Gårding: “Lacunas for hyperbolic differential operators with constant coefficients, II,” Acta Math. 131 : 1 (December 1973), pp. 145–206. A Russian translation was published in Usp. Mat. Nauk 39:3(237). MR 0470500 Zbl 0266.35045 article

Lars Gårding	Related
M. F. Atiyah	Related	Volume

R. Bott: “On the Chern–Weil homomorphism and the continuous cohomology of Lie-groups,” Adv. Math. 11 : 3 (December 1973), pp. 289–303. MR 0345115 Zbl 0276.55011 article

M. Atiyah, R. Bott, and V. K. Patodi: “Errata to: ‘On the heat equation and the index theorem’,” Invent. Math. 28 : 3 (1975), pp. 277–280. Errata for article in Invent. Math. 19:4 (1973). MR 0650829 Zbl 0301.58018 article

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

R. Bott: “On the Gelfand–Fuks cohomology,” pp. 357–364 in Differential geometry (Stanford, CA, 30 July–17 August 1973), part 1. Edited by S.-S. Chern and R. Osserman. Proceedings of Symposia in Pure Mathematics 27. American Mathematical Society (Providence, RI), 1975. MR 0380820 Zbl 0324.57013 incollection

Roughly four years ago Gelfand and Fuks had the inspired idea of considering the continuous cohomology of the Lie-algebra of \( C^{\infty} \)-vector fields on a manifold ([1969, 1970a, 1970b]). They showed that the answer was finite dimensional for all compact manifodls and that for \( S^1 \) this cohomology is a tensor product of an exterior algebra in dimension 3 and a polynomial ring with generator in dimension 2: \[ H_c(\mathcal{L}S^1) = E(\omega)\otimes \mathbf{R}[y]. \]

In this report I would like to present a new point of view towards their work which arose, on the one hand, out of the new insights into cohomology due to Guillemin and Trauber, and on the other out of conversations between G. Segal, A. Haefliger and myself during the past year. I will finally formulate a conjecture which would completely settle the homotopy theoretic nature of this cohomology. Although one finds evidence for its validity on every side, its proof has so far eluded us. I should also mention that when I recently showed this conjecture to Gelfand he told me that it had also, quite independently, been formulated by Fuks.

S. S. Chern	Related	Volume
Robert Osserman	Related

R. Bott: “On the shape of a curve,” Adv. Math. 16 : 2 (May 1975), pp. 144–159. Dedicated to Stan Ulam. Also published in Surveys in applied mathematics: Essays dedicated to Stanislaw Ulam (1976). MR 0383392 Zbl 0308.55011 article

R. Bott: “Some remarks on continuous cohomology,” pp. 161–170 in Manifolds-1973: Proceedings of the international conference on manifolds and related topics in topology (Tokyo, 10–17 April 1973). Edited by A. Hattori. University of Tokyo Press, 1975. MR 0368027 Zbl 0306.57013 incollection

R. Bott: “On the shape of a curve,” pp. 23–38 in Surveys in applied mathematics: Essays dedicated to Stanislaw Ulam (Los Alamos, NM, 12–18 June 1974). Edited by N. Poincaré, S. Orszag, and G.-C. Rota. Academic Press (New York), 1976. Also published in Adv. Math. 16:2 (1975). MR 0645193 incollection

Gian-Carlo Rota	Related
Steven Alan Orszag	Related
Stanisław Marcin Ulam	Related

R. Bott, H. Shulman, and J. Stasheff: “On the de Rham theory of certain classifying spaces,” Adv. Math. 20 : 1 (1976), pp. 43–56. MR 0402769 Zbl 0342.57016 article

Herbert Shulman	Related
James D. Stasheff	Related

R. Bott: “On characteristic classes in the framework of Gelfand–Fuks cohomology,” pp. 113–139 in Colloque “analyse et topologie” en l’honneur de Henri Cartan [Colloquium on analysis and topology in honor of Henri Cartan] (Orsay, France, 17–20 June 1974). Astérisque 32–33. Société Mathématique de France (Paris), 1976. MR 0402768 Zbl 0328.57005 incollection

R. Bott: “On the characteristic classes of groups of diffeomorphisms,” Enseign. Math., II. Sér. 23 : 3–4 (1977), pp. 209–220. Presented at the Colloquium on Topology and Algebra, Zürich, April 1977. MR 0488080 Zbl 0367.57004 article

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: “On some formulas for the characteristic classes of group-actions,” pp. 25–61 in Differential topology, foliations and Gelfand–Fuks cohomology (Rio de Janeiro, 5–24 January 1976). Edited by P. A. Schweitzer. Lecture Notes in Mathematics 652. Springer (Berlin), 1978. With an appendix by Robert Brooks. MR 505649 Zbl 0393.57011 incollection

Paul A. Schweitzer	Related
Robert Brooks	Related

R. Bott and R. Seeley: “Some remarks on the paper of Callias: ‘Axial anomalies and index theorems on open spaces’,” Commun. Math. Phys. 62 : 3 (1978), pp. 235–245. MR 507781 Zbl 0409.58019 article

Robert Thomas Seeley	Related
Constantine John Callias	Related

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: “The geometry and representation theory of compact Lie groups,” pp. 65–90 in Representation theory of Lie groups: Proceedings of the SRC/LMS research symposium on representations of Lie groups (Oxford, 28 June–15 July 1977). Edited by G. L. Luke. London Mathematical Society Lecture Note Series 34. Cambridge University Press, 1979. Notes by G. L. Luke. Zbl 0435.22015 incollection

R. Bott: “Morse theory and the Yang–Mills equations,” pp. 269–275 in Differential geometrical methods in mathematical physics (3–7 September 1979, Aix-en-Provence, France and 10–14 September 1979, Salamanca, Spain). Edited by P. L. García, A. Pérez-Rendón, and J. M. Souriau. Lecture Notes in Mathematics 836. Springer (Berlin), 1980. MR 607701 Zbl 0486.58009 incollection

In the variational problems of mathematical physics one is usually interested only in the extremals which afford an absolute minimum to the functional being varied. The other, the unstable solutions, therefore tend to be forgotten. On the other hand, in the Morse theory the totality of the extrema satisfy relations which are forced by the topology of the space underlying the variational problem. There is thus a certain amount of feedback between the stable and unstable critical sets which enables one to derive information about the topology of the absolute minimum by studying the sometimes more accessible unstable critical sets. The Yang–Mills functional for a principal bundle \( P \) over a compact Riemann surface \( M \) gives a striking illustration of the principle, and the purpose of this talk is to sketch in the main steps of this application of the Morse theory for you.

I am reporting on joint work with M. Atiyah still very much in progress.

A. Pérez-Rendón	Related
P. L. García	Related
Jean-Marie Souriau	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

R. Bott: “Equivariant Morse theory and the Yang–Mills equation on Riemann surfaces,” pp. 11–22 in The Chern symposium, 1979: Proceedings of the international symposium on differential geometry in honor of S.-S. Chern (Berkeley, CA, June 1979). Edited by W.-Y. Hsiang, S. Kobayashi, I. M. Singer, A. Weinstein, J. Wolf, and H.-H. Wu. Springer (New York), 1980. MR 609555 Zbl 0492.58011 incollection

Joseph Albert Wolf	Related
S. S. Chern	Related	Volume
Wu-yi Hsiang	Related
Shōshichi Kobayashi	Related	Volume
Alan Weinstein	Related
Isadore M. Singer	Related	Volume
Hung-hsi Wu	Related

R. Bott: “Marston Morse and his mathematical works,” Bull. Am. Math. Soc., New Ser. 3 : 3 (November 1980), pp. 907–950. MR 585177 Zbl 0469.01012 article

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

M. F. Atiyah and R. Bott: “Yang–Mills and bundles over algebraic curves,” pp. 11–20 in Geometry and analysis: Papers dedicated to the memory of V. K. Patodi, published as Proc. Indian Acad. Sci., Math. Sci. 90 : 1. Indian Academy of Sciences (Bangalore), 1981. MR 653942 Zbl 0499.14005 incollection

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

R. Bott: “Lectures on Morse theory, old and new,” Bull. Am. Math. Soc., New Ser. 7 : 2 (September 1982), pp. 331–358. Based on a paper in Proceedings of the 1980 Beijing symposium on differential geometry and differential equations (1982). An expanded and altered version of the paper in The mathematical heritage of Henri Poincaré (1983). MR 663786 Zbl 0505.58001 article

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

R. Bott: “Lectures on Morse theory, old and new,” pp. 169–218 in Proceedings of the 1980 Beijing symposium on differential geometry and differential equations (Beijing, 1980), vol. 1. Edited by S. S. Chern and W.-T. Wu. Science Press (Beijing), 1982. See also Bull. Am. Math. Soc. 7:2 (1982) and The mathematical heritage of Henri Poincaré (1983). MR 714336 Zbl 0521.58019 incollection

Wen-tsün Wu	Related
S. S. Chern	Related	Volume

R. Bott: “An equivariant setting of the Morse theory,” pp. 67–74 in Logic and algorithmic: An international symposium held in honour of Ernst Specker (Zürich, 5–11 February 1980). Edited by E. Engeler, H. Läuchli, and V. Strassen. Monographie de l’Enseignement Mathématique 30. University of Geneva, 1982. Originally published in Enseign. Math., II. Sér. 26:3–4 (1980). MR 648296 Zbl 0483.57019 incollection

Hans Läuchli	Related
Volker Strassen	Related
Erwin Engeler	Related
Ernst P. Specker	Related

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

R. Bott: “Lectures on Morse theory, old and new,” pp. 3–30 in The mathematical heritage of Henri Poincaré (Bloomington, IN, 7–10 April 1980), part 2. Edited by F. E. Browder. Proceedings of Symposia in Pure Mathematics 39. 1983. An expanded and altered version of this lecture appeared in Bull. Am. Math. Soc. 7:2 (1982). See also the paper in Proceedings of the 1980 Beijing symposium on differential geometry and differential equations (1982). Zbl 0525.58011 incollection

Felix E. Browder	Related
Jules Henri Poincaré	Related

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

M. F. At’ya, R. Bott, and L. Gårding: “Lacunas for hyperbolic differential operators with constant coefficients, II,” Usp. Mat. Nauk 39 : 3(237) (1984), pp. 171–224. Russian translation of an article in Acta Math. 131:1 (1973). MR 747794 Zbl 0568.35058 article

Lars Gårding	Related
M. F. Atiyah	Related	Volume

R. Bott: “On topology and other things,” Notices Am. Math. Soc. 32 : 2 (1985), pp. 152–158. MR 779225 article

R. Bott: “On some recent interactions between mathematics and physics,” Can. Math. Bull. 28 : 2 (1985), pp. 129–164. Jeffrey Williams lecture delivered at Canadian Mathematical Society meeting, Vancouver, 2 June 1983. MR 782770 Zbl 0569.58019 article

R. Bott: “Some aspects of homotopy and de Rham theory,” pp. 1–38 in Physics in higher dimensions (Jerusalem, 27 December 1984–4 January 1985). Edited by T. Piran and S. Weinberg. Jerusalem Winter School for Theoretical Physics 2. World Scientific (Singapore), 1986. MR 875029 incollection

Steven Weinberg	Related
Tsvi Piran	Related

R. Bott: “On characteristic classes and continuous cohomology,” pp. 29–40 in Supersymmetry and its applications: Superstrings, anomalies and supergravity (Cambridge, UK, 23 June–14 July 1985). Edited by G. W. Gibbons, S. W. Hawking, and P. K. Townsend. Cambridge University Press, 1986. MR 862800 Zbl 0609.57011 incollection

Gary William Gibbons	Related
Paul Kingsley Townsend	Related
Stephen William Hawking	Related

R. Bott: “On the fixed point formula and the rigidity theorems of Witten lectures at Cargèse 1987,” pp. 13–32 in Nonperturbative quantum field theory (Cargèse, Corsica, 16–30 July 1987). Edited by G. ’t Hooft, A. Jaffe, G. Mack, P. K. Mitter, and R. Stora. NATO Advanced Science Institutes Series B: Physics 185. Plenum (New York), 1988. MR 1008274 incollection

Pronob K. Mitter	Related
Gerhard Mack	Related
Arthur Jaffe	Related
Raymond Stora	Related
Gerardus ’t Hooft	Related

R. Bott: “Morse theory indomitable,” Publ. Math., Inst. Hautes Étud. Sci. 68 (1988), pp. 99–114. Dedicated to René Thom. MR 1001450 Zbl 0685.58009 article

R. Bott: “On induced representations,” pp. 1–13 in The mathematical heritage of Hermann Weyl (Durham, NC, 12–16 May 1987). Edited by R. O. Wells, Jr. Proceedings of Symposia in Pure Mathematics 48. American Mathematical Society (Providence, RI), 1988. A Russian translation was published in Usp. Mat. Nauk 45:3(273) (1990). MR 974328 Zbl 0657.22020 incollection

Hermann Weyl	Related
Raymond O'Neil Wells, Jr	Related

R. H. Bott: The topological constraints on analysis, 1989. 60 minute videocassette. AMS-MAA Joint Lecture Series. American Mathematical Society (Providence, RI). Gleason introduces Bott. Recorded in Providence, RI, 9 August 1988. MR 1057176 Zbl 0925.01018 misc

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 and L. V. Tu: Differentsial’nye formy v algebraicheskoj topologii [Differential forms in algebraic topology]. Nauka (Moscow), 1989. Russian translation of Differential forms in algebraic topology (1982). MR 1085026 Zbl 0675.55002 book

Loring W. Tu	Related
Alexandre Aleksandrovich Kirillov	Related

R. Bott: “The topological constraints on analysis,” pp. 527–542 in A century of mathematics in America, part II. Edited by P. L. Duren, R. Askey, and U. C. Merzbach. History of Mathematics 2. American Mathematical Society (Providence, RI), 1989. MR 1003154 Zbl 0666.01012 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Richard Allen Askey	Related	Volume

R. Bott: “On induced representations,” Usp. Mat. Nauk 45 : 3(273) (1990), pp. 85–96. Russian translation of an article in Proc. Symp. Pure Math. 48 (1988). MR 1071935 Zbl 0714.22014 article

R. Bott: “Georges de Rham: 1901–1990,” Notices Am. Math. Soc. 38 : 2 (1991), pp. 114–115. MR 1090924 Zbl 1194.01069 article

R. Bott: “Stable bundles revisited,” pp. 1–18 in Proceedings of the conference on geometry and topology (Cambridge, MA, 27–29 April 1990). Edited by H. B. Lawson, Jr. and S.-T. Yau. Surveys in Differential Geometry 1. International Press (Sommerville, MA), 1991. Conference sponsored by Lehigh University. MR 1144525 Zbl 0757.32013 incollection

H. Blaine Lawson, Jr.	Related
Shing-Tung Yau	Related

R. Bott: “On E. Verlinde’s formula in the context of stable bundles,” pp. 2847–2858 in Topological methods in quantum field theory (Trieste, Italy, 11–15 June 1990), published as Int. J. Mod. Phys. A 6 : 16. Issue edited by W. Nahm, S. Randjbar-Daemi, E. Sezgin, and E. Witten. 1991. MR 1117752 Zbl 0820.14024 incollection

Werner Nahm	Related
S. Randjbar-Daemi	Related
E. Sezgin	Related
Edward Witten	Related

R. Bott: “For the Chern volume,” pp. 106–108 in S. S. Chern: A great geometer of the twentieth century (Los Angeles, 1990). Edited by S.-T. Yau. International Press (Cambridge, MA), 1992. MR 1201350 incollection

S. S. Chern	Related	Volume
Shing-Tung Yau	Related

R. Bott: “On knot and manifold invariants,” pp. 37–52 in New symmetry principles in quantum field theory (Cargèse, Corsica, 16–27 July 1991). Edited by J. Fröhlich, G. ’t Hooft, A. Jaffe, G. Mack, P. K. Mitter, and R. Stora. NATO Advanced Science Institutes Series B: Physics 295. Plenum (New York), 1992. MR 1204451 Zbl 1221.57002 incollection

Pronob K. Mitter	Related
Gerhard Mack	Related
Arthur Jaffe	Related
Josef Fröhlich	Related
Raymond Stora	Related
Gerardus ’t Hooft	Related

R. Bott: “Topological aspects of loop groups,” pp. 65–80 in Topological quantum field theories and geometry of loop spaces (Budapest, 25 June–1 July 1989). Edited by L. Fehér, A. Stipsicz, and J. Szenthe. World Scientific (Singapore), 1992. MR 1260751 incollection

János Szenthe	Related
András Stipsicz	Related
László Fehér	Related

R. Bott: “Reflections on the theme of the poster,” pp. 125–135 in Topological methods in modern mathematics: A symposium in honor of John Milnor’s sixtieth birthday (Stony Brook, NY, 14–21 June 1991). Edited by L. R. Goldberg and A. V. Phillips. Publish or Perish (Houston, TX), 1993. MR 1215962 Zbl 0817.57003 incollection

Lisa Robin Goldberg	Related
Anthony Valiant Phillips	Related
John W. Milnor	Related

R. Bott: “Luncheon talk and nomination for Stephen Smale,” pp. 67–72 in From topology to computation: Proceedings of the Smalefest (Berkeley, CA, 5–9 August 1990). Edited by M. W. Hirsch, J. E. Marsden, and M. Shub. Springer (New York), 1993. MR 1246108 Zbl 0795.01016 incollection

Jerrold E. Marsden	Related
Michael Ira Shub	Related
Stephen Smale	Related
Morris W. Hirsch	Related

M. Atiyah: “Contribution to the collected works of Raoul Bott,” pp. xxix–xxx in Raoul Bott: Collected papers, vol. 2: Differential operators. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1994. MR 1290365 incollection

M. F. Atiyah	Related	Volume
Robert Duncan MacPherson	Related

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

S. Smale: “Some early recollection of Raoul Bott,” pp. i–xii in Raoul Bott: Collected papers, vol. 1: Topology and Lie groups. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1994. MR 1280037 incollection

Stephen Smale	Related
Robert Duncan MacPherson	Related

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: “Comments on the papers in Volume 1,” pp. 27–35 in Collected papers, vol. 1: Topology and Lie groups. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1994. MR 1280035 incollection

R. Bott and C. Taubes: “On the self-linking of knots,” pp. 5247–5287 in Topology and physics, published as J. Math. Phys. 35 : 10 (1994). MR 1295465 Zbl 0863.57004 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

E. Getzler: “The Atiyah–Bott fixed point formula,” pp. xxxi–xxxiii in Raoul Bott: Collected papers, vol. 2: Differential operators. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1994. MR 1290366 incollection

Ezra Getzler	Related
Robert Duncan MacPherson	Related

R. Bott: “Comments on some of the papers in Volume 2,” pp. xvii–xxii in Collected papers, vol. 2: Differential operators. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston, MA), 1994. MR 1290363 incollection

Current developments in mathematics, 1995 (Boston, MA, 7–8 May 1995). Edited by R. Bott, A. Jaffe, S.-T. Yau, M. Hopkins, I. Singer, and D. Stroock. Current Developments in Mathematics. International Press (Cambridge, MA), 1995. MR 1474976 Zbl 0833.00016 book

Michael Jerome Hopkins	Related
Daniel W. Stroock	Related
Arthur Jaffe	Related
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume

C. Taubes: “Appreciation for Raoul Bott and his mathematics/physics career,” pp. xv–xvii in Raoul Bott: Collected papers, vol. 4: Mathematics related to physics. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1995. MR 1321891 incollection

Robert Duncan MacPherson	Related
Clifford Henry Taubes	Related

R. Bott: “Report on the fixed point formula,” pp. 265–270 in Séminaire Bourbaki 9 (1964–1966). Société Mathématique de France (Paris), 1995. Exposé no. 295. MR 1608803 incollection

“Raoul Bott,” pp. 229–230 in Notable twentieth century scientists, vol. 1. Edited by E. J. MacMurray, J. K. Kosek, and R. M. Valade. Gale Research (New York), 1995. incollection

Emily J. MacMurray	Related
Jane Kelly Kosek	Related
Roger M. Valade	Related

A. Haefliger: “Raoul Bott and foliation theory in the 1970s,” pp. xxvii–xxxi in Raoul Bott: Collected papers, vol. 3: Foliations. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1995. MR 1321889 incollection

André Haefliger	Related
Robert Duncan MacPherson	Related

P. Baum: “Working with Bott (rocking and rolling with Raoul),” pp. xxii–xxiii in Raoul Bott: Collected papers, vol. 3: Foliations. Edited by R. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1995. MR 1321887 incollection

Paul Baum	Related	Volume
Robert Duncan MacPherson	Related

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

L. Conlon: “Raoul Bott, foliations, and characteristic classes. An appreciation,” pp. xxiv–xvi in Raoul Bott: Collected papers, vol. 3: Foliations. Edited by R. D. MacPherson. Contemporary Mathematicians. Birkhäuser (Boston), 1995. MR 1321888 incollection

Robert Duncan MacPherson	Related
Lawrence William Conlon	Related

R. Bott: “On invariants of manifolds,” pp. 29–39 in Modern methods in complex analysis (Princeton, NJ, 16–20 March 1992). Edited by T. Bloom, D. Catlin, J. P. D’Angelo, and Y.-T. Siu. Annals of Mathematics Studies 137. Princeton University Press, 1995. Conference in honor of Robert Clifford Gunning and John Joseph Kohn. MR 1369132 Zbl 0846.57016 incollection

Thomas Bloom	Related
David William Catlin	Related
John P. D’Angelo	Related
Yum-Tong Siu	Related
Robert Clifford Gunning	Related
Joseph John Kohn	Related

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

R. Bott: “Configuration spaces and imbedding invariants,” pp. 1–17 in Proceedings of the 4th Gökova geometry-topology conference (Gökova, Turkey, 29 May–2 June 1995), published as Turkish J. Math. 20 : 1. Issue edited by S. Akbulut, T. Önder, and R. J. Stern. 1996. MR 1392659 Zbl 0880.57002 incollection

Turgut Önder	Related
Selman Akbulut	Related
Ronald John Stern	Related

Current developments in mathematics 1996 (Cambridge, MA, 1996). Edited by R. Bott, A. Jaffe, D. Jerison, G. Lusztig, I. Singer, and S.-T. Yau. Current Developments in Mathematics. International Press (Cambridge, MA), 1997. MR 1724942 Zbl 0899.00015 book

George Lusztig	Related
Arthur Jaffe	Related
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume
David Saul Jerison	Related

R. Bott: “Critical point theory in mathematics and in mathematical physics,” pp. 9–40 in Proceedings of the 5th Gökova geometry-topology conference (Gökova, Turkey, 27–31 May 1996), published as Turk. J. Math. 21 : 1. Issue edited by S. Akbulut, T. Önder, and R. J. Stern. 1997. MR 1456156 Zbl 0889.58021 incollection

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 Alexrod and Singer [1994] and Kontsevich [1992] in their work on 3-manifold invariants.

This year I thought I would comment on how these invariants arise out of physics-inspired considerations, but in a manner which might be more palatable to mathematicians than other accounts I can point to in the literature.

At the heart of the matter, as I see it, is the quite different role critical point theory plays in topology and in quantum physics, and I will therefore start with a comparison of these roles in the simplest instance: that of a compact oriented manifold \( M \) equipped with a “Morse function” \( f:M\to R \).

Turgut Önder	Related
Selman Akbulut	Related
Ronald John Stern	Related

R. Bott: “Configuration spaces and imbedding problems,” pp. 135–140 in Geometry and physics (Aarhus, Denmark, 1995). Edited by J. E. Andersen, J. Dupont, H. Pedersen, and A. Swann. Lecture Notes in Pure and Applied Mathematics 184. Dekker (New York), 1997. MR 1423159 Zbl 0874.57006 incollection

Johan L. Dupont	Related
Jørgen Ellegaard Andersen	Related
Henrik Pedersen	Related
Andrew Swann	Related

R. Bott and A. S. Cattaneo: “Integral invariants of 3-manifolds,” J. Differ. Geom. 48 : 1 (1998), pp. 91–133. MR 1622602 Zbl 0953.57008 article

R. Bott, C. Earle, D. Hejhal, J. Jenkins, T. Jorgensen, A. Marden, and R. Osserman: “Lars Valerian Ahlfors (1907–1996),” Notices Am. Math. Soc. 45 : 2 (1998), pp. 248–255. Edited and with an introduction by Steven G. Krantz. MR 1601820 Zbl 0908.01015 article

Clifford John Earle, Jr.	Related
Lars Valerian Ahlfors	Related
Robert Osserman	Related
Steven Krantz	Related
Dennis Arnold Hejhal	Related
James Jenkins	Related
Troels Jørgensen	Related

Current developments in mathematics 1997 (Cambridge, MA, 1997). Edited by R. Bott, A. Jaffe, S.-T. Yau, D. Jerison, G. Lusztig, and I. Singer. Current Developments in Mathematics. International Press (Boston), 1999. A preliminary version was published in 1997. MR 1698851 Zbl 0927.00019 book

George Lusztig	Related
Arthur Jaffe	Related
Shing-Tung Yau	Related
Isadore M. Singer	Related	Volume
David Saul Jerison	Related

R. Bott: “A letter from Raoul Bott,” Asian J. Math. 3 : 1 (1999), pp. xli–xlii. Zbl 0958.01020 article

R. Bott and A. S. Cattaneo: “Integral invariants of 3-manifolds, II,” J. Differ. Geom. 53 : 1 (1999), pp. 1–13. MR 1776090 Zbl 1036.57500 article

R. Bott: “An introduction to equivariant cohomology,” pp. 35–56 in Quantum field theory: Perspective and prospective (Les Houches, France, 15–26 June 1998). Edited by C. DeWitt-Morette and J.-B. Zuber. NATO Science Series C: Mathematical and Physical Sciences 530. Kluwer Academic (Dordrecht), 1999. MR 1725010 Zbl 0970.55003 incollection

Jean-Bernard Zuber	Related
Cécile M. DeWitt-Morette	Related

R. Bott and L. W. Tu: “Equivariant characteristic classes in the Cartan model,” pp. 3–20 in Proceedings of the international conference on geometry, analysis and applications (Varanasi, India, 21–24 August 2000). Edited by R. S. Pathak. World Scientific (Singapore), 2001. Dedicated to the memory of V. K. Patodi. MR 1850463 Zbl 1016.53024 incollection

R. S. Pathak	Related
Vijay Kumar Patodi	Related
Loring W. Tu	Related

R. Bott and C. Taubes: “A remark on integral geometry,” pp. 59–83 in Essays on geometry and related topics: Mémoires dédiés à André Haefliger [Essays on geometry and related topics: Memoirs dedicated to André Haefliger], vol. 1. Edited by É. Ghys. Monographie de l’Enseignement Mathématique 38. University of Geneva, 2001. MR 1929322 Zbl 1021.53049 incollection

André Haefliger	Related
Clifford Henry Taubes	Related
Étienne Ghys	Related

A. Jackson: “Interview with Raoul Bott,” Notices Am. Math. Soc. 48 : 4 (2001), pp. 374–382. The edited text of two interviews with Raoul Bott conducted in October 2000. Republished in The founders of index theory (2003). MR 1816297 Zbl 1047.01523 article

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
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A. Jackson: “Interview with Raoul Bott,” pp. 101–114 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2003. The edited text of two interviews with Raoul Bott conducted in October 2000. Republished from Notices Am. Math. Soc. 48:4 (2001). incollection

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R. Bott, S. Tolman, and J. Weitsman: “Surjectivity for Hamiltonian loop group spaces,” Invent. Math. 155 : 2 (2004), pp. 225–251. MR 2031427 Zbl 1067.53067 article

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R. Bott: “Introduction,” pp. vii–xi in Woods Hole mathematics: Perspectives in mathematics and physics (Woods Hole, MA). Edited by N. Tongring and R. C. Penner. Series on Knots and Everything 34. World Scientific (Singapore), 2004. MR 2123365 incollection

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J. Pearce: “Raoul Bott, an innovator in mathematics, dies at 82,” The New York Times (8 January 2006). article

L. W. Tu: “The life and works of Raoul Bott,” Notices Am. Math. Soc. 53 : 5 (2006), pp. 554–570. Updated reprint of an article published in The founders of index theory (2003). MR 2254401 Zbl 1142.01352 article

M. Atiyah: “Raoul Harry Bott: 24 September 1923–20 December 2005,” Biogr. Mems Fell. R. Soc. 53 (December 2007), pp. 63–76. Republished in Bull. Lond. Math. Soc. 42:1 (2010). article

The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer, 2nd edition. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. Republication of 2003 original. MR 2547480 Zbl 1195.01090 book

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“For Raoul,” pp. 95–100 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. Authored anonymously by “a friend of Raoul”. incollection

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P. D. Lax, F. Hirzebruch, B. Mazur, L. Conlon, E. B. Curtis, H. M. Edwards, J. Huebschmann, and H. Shulman: “Raoul Bott as we knew him,” pp. 43–49 in A celebration of the mathematical legacy of Raoul Bott (Montreal, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2648884 Zbl 1195.01029 incollection

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J. Lambek: “Raoul Bott, McGill, the 1940s,” pp. 13–15 in A celebration of the mathematical legacy of Raoul Bott (Montréal, QC, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2648902 Zbl 1195.01069 incollection

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J. A. Bernhard, N. Hingston, J. Stasheff, and V. Guillemin: “Raoul Bott as we knew him,” pp. 153–159 in A celebration of the mathematical legacy of Raoul Bott (Montréal, QC, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2648893 Zbl 1195.01023 incollection

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J. H. Hubbard: “The Bott–Duffin synthesis of electrical circuits,” pp. 33–40 in A celebration of the mathematical legacy of Raoul Bott (Montréal, QC, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2648883 Zbl 1201.94146 incollection

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S. Lu: “A physics colloquium at McGill that changed my life,” pp. 279 in A celebration of the mathematical legacy of Raoul Bott (Montréal, QC, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2640504 Zbl 1195.01031 incollection

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R. Bott: Collected papers, vol. 5. Edited by E. by L. W. Tu. Contemporary Mathematicians. Springer/Birkhäuser, Cham (Berlin), 2017. MR 3848672 book

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