Celebratio Mathematica — Singer — Complete Bibliography

[1] I. M. Singer: Lie algebras of unbounded operators. Ph.D. thesis, University of Chicago, June 1950. Advised by I. E. Segal. MR 2611383 phdthesis

[2] R. V. Kadison and I. M. Singer: “Some remarks on representations of connected groups,” Proc. Nat. Acad. Sci. U. S. A. 38 : 5 (May 1952), pp. 419–423. MR 48456 Zbl 0046.25202 article

The purpose of this note is to bring to light a fact which has escaped notice, viz., in the direct integral reduction of the regular representation of a connected separable locally compact group, factors of Type \( \mathrm{II}_1 \) occur almost nowhere. This proof is carried out by the following scheme of argument. We show first that a connected locally compact group which has sufficiently many unitary representations which generate rings of finite type is the group direct product of a compact group and an abelian group. From this it follows quite easily that a unitary representation of a connected locally compact group generates a ring of operators which has no summand of Type \( \mathrm{II}_1 \) and, in particular, is not itself a factor of Type \( \mathrm{II}_1 \). Employing a theorem of Mautner, to the effect that, for almost every factor in the direct integral reduction of the regular representation of a group, there exists a strongly continuous representation of the group which generates the factor, we obtain the final result.

[3] I. M. Singer: “Uniformly continuous representations of Lie groups,” Ann. Math. (2) 56 : 2 (September 1952), pp. 242–247. MR 49201 Zbl 0049.35802 article

It is known that noncompact semisimple and nonabelian solvable Lie groups do not have faithful finite-dimensional unitary (and therefore bounded) representations. In fact if none of the simple components of a semisimple Lie group are compact, all such representations are trivial. This amounts to minimal almost periodicity for such groups [von Neumann and Segal 1950]. On the other hand, any locally compact group has a faithful infinite dimensional unitary representation, the regular representation, which is continuous in the strong operator topology on the group of unitary operators on a Hilbert space. In [Gelfand and Raikov 1943], it is shown that there are sufficiently many irreducible unitary representations of a locally compact group continuous in the strong operator topology.

However, if we restrict our attention to unitary representations continuous in the uniform operator topology, the finite dimensional situation obtains and it is the object of this paper to prove these results (see the corollaries following Theorem 2). Since the uniform and strong operator topologies coincide for finite dimensional spaces, the proof covers the non-existence of finite dimensional bounded representations as well.

[4] W. Ambrose and I. M. Singer: “A theorem on holonomy,” Trans. Amer. Math. Soc. 75 : 3 (1953), pp. 428–443. MR 63739 Zbl 0052.18002 article

The object of this paper is to prove Theorem 2 of §2, which shows, for any connexion, how the curvature form generates the holonomy group. We believe this is an extension of a theorem stated without proof by E. Cartan [1926, p.4]. This theorem was proved after we had been informed of an unpublished related theorem of Chevalley and Koszul. We are indebted to S.-S. Chern for many discussions of matters considered here.

In §1 we give an exposition of some needed facts about connexions; this exposition is derived largely from an exposition of Chern [1951] and partly from expositions of H. Cartan [1951] and Ehresmann [1951]. We believe this exposition does however contain one new element, namely Lemma 1 of §1 and its use in passing from H. Cartan’s definition of a connexion (the definition given in §1) to E. Cartan’s structural equation.

[5] R. Arens and I. M. Singer: “Function values as boundary integrals,” Proc. Amer. Math. Soc. 5 : 5 (1954), pp. 735–745. MR 65024 Zbl 0056.33501 article

[6] I. M. Singer: “Automorphisms of finite factors,” Amer. J. Math. 77 : 1 (January 1955), pp. 117–133. MR 66567 Zbl 0064.11001 article

[7] I. M. Singer and J. Wermer: “Derivations on commutative normed algebras,” Math. Ann. 129 : 1 (1955), pp. 260–264. MR 70061 Zbl 0067.35101 article

[8] I. M. Singer: “Report on group representations,” pp. 11–26 in Report of an international conference on operator theory and group representations (Harriman, NY, 20–23 October 1953). NRC Publications 387. National Academy of Sciences and National Research Council (Washington DC), 1955. MR 76285 incollection

[9] R. Arens and I. M. Singer: “Generalized analytic functions,” Trans. Amer. Math. Soc. 81 (2 1956), pp. 379–393. MR 78657 Zbl 0078.10902 article

[10] K. Hoffman and I. M. Singer: “Maximal subalgebras of \( C(\Gamma) \),” Amer. J. Math. 79 : 2 (April 1957), pp. 295–305. MR 85478 Zbl 0078.11001 article

[11] R. V. Kadison and I. M. Singer: “Three test problems in operator theory,” Pac. J. Math. 7 : 2 (February 1957), pp. 1101–1106. MR 92123 Zbl 0078.11503 article

[12] I. M. Singer: “The problem of homeomorphism of separable infinite-dimensional Banach spaces,” Acad. Republ. Popul. Romîne, An. romîne-sovietice, Ser. Mat.-Fiz. (3) 12 : 4(27) (1958), pp. 5–24. MR 102726 Zbl 0085.31902 article

[13] W. Ambrose and I. M. Singer: “On homogeneous Riemannian manifolds,” Duke Math. J. 25 : 4 (1958), pp. 647–669. MR 102842 Zbl 0134.17802 article

[14] I. M. Singer: “The geometric interpretation of a special connection,” Pac. J. Math. 9 : 2 (June 1959), pp. 585–590. MR 111062 Zbl 0086.15101 article

An esthetic problem remaining in the foundations of Riemannian geometry is this. To each Riemannian metric on a manifold one associates a connection on its bundle of bases. This connection is uniquely characterized by its having zero torsion and its parallel translation preserving the metric. By very definition at each tangent space of the bundle the connection assigns a complement to the vertical. Yet the various existence proofs of the Riemannian connection are computational and do not exhibit this complementary space (the horizontal space) directly. The problem is how can one do so directly from the metric, given (say) by the reduction of the bundle to the bundle of frames.

Under rather special circumstances, we exhibit a geometric and coordinate free construction of a connection in §2. This includes the class of complex analytic bundles and gives a geometric construction of a connection of type \( (1,0) \) with curvature of type \( (1,1) \), useful in algebraic geometry (§4). See [Atiyah 1957] and [Kodaira and Spencer 1958]. For the case of complex vector bundles Nakano computes this connection in terms of coordinates [Nakano 1955]. For Kähler manifolds it turns out (§2) that this connection is the Riemannian connection restricted to the unitary subbundle. Hence we obtain a solution to the problem posed in the preceding paragraph for Kähler manifolds.

[15] K. Goffman and I. M. Zinger: “Some problems of Gel’fand,” Uspehi Mat. Nauk 14 : 3(87) (1959), pp. 99–114. An English version was published in Eighteen papers on algebra (1963). MR 117538 Zbl 0154.38502 article

Kenneth Hoffman	Related
Israïl Moiseevich Gelfand	Related

[16] R. V. Kadison and I. M. Singer: “Extensions of pure states,” Amer. J. Math. 81 : 2 (April 1959), pp. 383–400. MR 123922 Zbl 0086.09704 article

The main concern of this paper is the problem of uniqueness of extensions of pure states from maximal abelian self-adjoint algebras of operators on a Hilbert space to the algebra of bounded operators on that space. The answer, as many of have suspected for several years, is in the negative. To the best of our knowledge, the problem is not recorded in the literature. We heard of it first from I. E. Segal and I. Kaplansky, though it is difficult to credit a problem which stems naturally from the physical interpretation and the inherent structure of a subject. This problem has arisen, in one form or another, in our work on several different occasions; and we have been gathering bits of information related to it, over the years. (The present solution is prompted by just such a reappearance.)

[17] K. Hoffman and I. M. Singer: “Maximal algebras of continuous functions,” Acta Math. 103 : 3–4 (1960), pp. 217–241. MR 117540 Zbl 0195.13903 article

[18] R. V. Kadison and I. M. Singer: “Triangular operator algebras: Fundamentals and hyperreducible theory,” Amer. J. Math. 82 : 2 (April 1960), pp. 227–259. MR 121675 Zbl 0096.31703 article

[19] W. Ambrose, R. S. Palais, and I. M. Singer: “Sprays,” An. Acad. Brasil. Ci. 32 : 2 (June 1960), pp. 163–178. MR 126234 Zbl 0097.37904 article

Richard Sheldon Palais	Related	Volume
Warren Arthur Ambrose	Related

[20] I. M. Singer: “Infinitesimally homogeneous spaces,” Comm. Pure Appl. Math. 13 : 4 (November 1960), pp. 685–697. MR 131248 Zbl 0171.42503 article

[21] I. Singer: “Basic sequences and reflexivity of Banach spaces,” Studia Math. 21 (1961–1962), pp. 351–369. A related paper was published in General topology and its relations to modern analysis and algebra (1962). MR 146635 article

[22] I. Singer: “Basic sequences and reflexivity of Banach spaces,” pp. 331–332 in General topology and its relations to modern analysis and algebra (Prague, September 1961). Academic Press (New York), 1962. A related article was published in Studia Math. 21 (1961–1962). Zbl 0114.30902 incollection

[23] K. Hoffman and I. M. Singer: “On some problems of Gel’fand,” pp. 143–157 in Eighteen papers on algebra. American Mathematical Society Translations. Series 2 27. 1963. Translation of Russian original published in Uspehi Mat. Nauk 14:3(87) (1959). MR 151866 Zbl 0187.38601 incollection

Kenneth Hoffman	Related
Israïl Moiseevich Gelfand	Related

[24] M. F. Atiyah and I. M. Singer: “The index of elliptic operators on compact manifolds,” Bull. Amer. Math. Soc. 69 : 3 (1963), pp. 422–433. MR 157392 Zbl 0118.31203 article

In [1960] Gel’fand posed the general problem of investigating the relationship between topological and analytical invariants of elliptic differential operators. In particular he suggested that it should be possible to express the index of an elliptic operator in topological terms. This problem has been taken up by Agranovič [1962a; 1962b], Dynin [1962b; 1961a; 1961b], Seeley [1961; 1963] and Vol’pert [1962] who have solved it in special cases. The purpose of this paper is to give a general formula for the index of an elliptic operator on any compact oriented differentiable manifold. As a special case of this formula we get the Hirzebruch–Riemann–Roch theorem for any compact complex manifold. This was previously known only for projective algebraic manifolds. Some other special cases, of interest in differential topology, are discussed.

[25] I. M. Singer and S. Sternberg: “The infinite groups of Lie and Cartan, I: The transitive groups,” J. Analyse Math. 15 : 1 (1965), pp. 1–114. MR 217822 Zbl 0277.58008 article

[26] V. Guillemin and I. M. Singer: “Differential equations and \( G \)-structures,” pp. 34–36 in Proceedings of the United States–Japan seminar on differential geometry (Kyoto, 14–19 June 1965). Edited by K. Nomizu, Y. Matsushima, and K. Yano. Nippon Hyoronsha (Tokyo), 1966. MR 213988 Zbl 0163.33402 incollection

Kentaro Yano	Related
Katsumi Nomizu	Related
Victor William Guillemin	Related
Yozô Matsushima	Related

[27] I. M. Singer and J. A. Thorpe: Lecture notes on elementary topology and geometry. Scott, Foresman and Co. (Glenview, IL), 1967. Reprinted in 1976. MR 213982 Zbl 0163.44302 book

[28] H. P. McKean, Jr. and I. M. Singer: “Curvature and the eigenvalues of the Laplacian,” J. Diff. Geom. 1 : 1 (1967), pp. 43–69. MR 217739 Zbl 0198.44301 article

[29] M. F. At’ja and I. M. Zinger: “The index of elliptic operators, I,” Uspehi Mat. Nauk 23 : 5(143) (1968), pp. 99–142. Russian translation of an article published in Ann. Math. (2) 87:3 (1968). MR 232402 article

[30] M. F. Atiyah and I. M. Singer: “The index of elliptic operators, I,” Ann. Math. (2) 87 : 3 (May 1968), pp. 484–530. A Russian translation was published in Uspehi Mat. Nauk 23:5(143). MR 236950 Zbl 0164.24001 article

This is the first of a series of papers which will be devoted to a study of the index of elliptic operators on compact manifolds. The main result was announced in [Atiyah and Singer 1963] and, for manifolds with boundary, in [Atiyah 1964]. The long delay between these announcements and the present paper is due to several factors. On the one hand, a fairly detailed exposition has already appeared in [Palais 1965]. On the other hand, our original proof, reproduced with minor modifications in [Palais 1965], had a number of drawbacks. In the first place the use of cobordism, and the computational checking associated with this, were not very enlightening. More seriously, however, the method of proof did not lend itself to certain natural generalizations of the problem where appropriate cobordism groups were not known. The reader who is familiar with the Riemann–Roch theorem will realize that our original proof of the index theorem was modelled closely on Hirzebruch’s proof of the Riemann–Roch theorem. Naturally enough we were led to look for a proof modelled more on that of Grothendieck. While we have not completely succeeded in this aim, we have at least found a proof which is much more natural, does not use cobordism, and lends itself therefore to generalization. In spirit, at least, it has much in common with Grothendieck’s approach.

[31] M. F. Atiyah and I. M. Singer: “The index of elliptic operators, III,” Ann. Math. (2) 87 : 3 (May 1968), pp. 546–604. A Russian translation was published in Uspehi Mat. Nauk 24:1(145). MR 236952 Zbl 0164.24301 article

In [1968a], paper I of this series, the index of an elliptic operator was computed in terms of \( K \)-theory. In this paper, we carry out what is essentially a routine exercise by passing from \( K \)-theory to cohomology. In this way, we end up with the explicit cohomological formula for the index announced in [1963].

In [1968a] we also considered elliptic operators (or complexes) compatible with a compact group \( G \) of transformations. The index in this case is a character of \( G \), and the main theorem of [1968a] gave a construction for this in \( KG \)-theory. In [1968b], paper II of this series, the value of this index-character at an element \( g\in G \) was expressed as the index of a new “virtual operator” on the fixed point set of \( g \). This was referred to as a Lefschetz fixed-point formula. By combining this formula with the cohomological formula for the index, we obtain finally an explicit cohomological formula for the index-character. We shall describe this formula in detail for a number of important operators. In particular we draw attention to the “integrality theorems” obtained in this way for actions of finite groups on manifolds. Most of these do not depend on the analysis in [1968a], but are a consequence of combining the purely topological results of [1968b] and the present paper.

[32] I. M. Singer: “Elliptic operators on manifolds,” pp. 333–375 in Pseudo-differential operators (Stresa, Italy, 26 August–3 September 1968). Edited by L. Nirenberg. C.I.M.E. Summer Schools 47. Edizioni Cremonese (Rome), 1968. MR 262712 incollection

[33] S. Doplicher, T. Regge, and I. M. Singer: “A geometrical model showing the independence of locality and positivity of the energy,” Commun. Math. Phys. 7 : 1 (1968), pp. 51–54. MR 1552528 Zbl 0179.58202 article

Tullio Regge	Related
Sergio Doplicher	Related

[34] I. M. Singer and J. A. Thorpe: “The curvature of 4-dimensional Einstein spaces,” pp. 355–365 in Global analysis: Papers in honor of K. Kodaira. Edited by D. C. Spencer and S. Iyanaga. Princeton University Press, 1969. MR 256303 Zbl 0199.25401 incollection

Kunihiko Kodaira	Related
Shōkichi Iyanaga	Related
Donald Clayton Spencer	Related
John Alden Thorpe	Related

[35] M. F. At’ja and I. M. Zinger: “The index of elliptic operators, III,” Uspehi Mat. Nauk 24 : 1(145) (1969), pp. 127–182. The English original was published in Ann. Math. (2) 87:3 (1968). MR 256417 article

[36] M. F. Atiyah and I. M. Singer: “Index theory for skew-adjoint Fredholm operators,” Inst. Hautes Études Sci. Publ. Math. 37 (1969), pp. 5–26. MR 285033 Zbl 0194.55503 article

[37] M. F. Atiyah and I. M. Singer: “The index of elliptic operators, IV,” Ann. Math. (2) 93 : 1 (January 1971), pp. 119–138. A Russian translation was published in Uspehi Mat. Nauk 27:4(166) (1972). MR 279833 Zbl 0212.28603 article

[38] D. B. Ray and I. M. Singer: “\( R \)-torsion and the Laplacian on Riemannian manifolds,” Adv. Math. 7 : 2 (October 1971), pp. 145–210. MR 295381 Zbl 0239.58014 article

Let \( W \) be a compact oriented Riemannian manifold of dimension \( N \), and let \( K \) be a simplicial complex which is a smooth triangulation of \( W \). The Reidemeister–Franz torsion (or \( R \)-torsion) \( \tau \) of \( K \) is a function of certain representations of the fundamental group of \( K \). Since it is a combinatorial invariant, and since smooth triangulations of \( W \) are equivalent, this torsion is a manifold invariant.

We raise the question as to how to describe this manifold invariant in analytic terms. Arnold Shapiro once suggested that there might be a formula for the torsion in terms of the Laplacian \( \Delta \) acting on differential forms on \( W \). Our candidate \( T \) involves the zeta function for appropriate Laplacians. Though we have been unable to prove that \( T = \tau \), we show in this paper that \( T \) is a manifold invariant and present some evidence that \( T = \tau \).

[39] F. Hirzebruch, L. Hörmander, J. Milnor, J.-P. Serre, and I. M. Singer: Prospects in mathematics. Annals of Mathematics Studies 70. Princeton University Press, 1971. Zbl 0233.00003 book

F. Hirzebruch	Related	Volume
Lars Hörmander	Related
John W. Milnor	Related
Jean-Pierre Serre	Related

[40] I. M. Singer: “Future extensions of index theory and elliptic operators,” pp. 171–185 in F. Hirzebruch, L. Hörmander, J. Milnor, J.-P. Serre, and I. M. Singer: Prospects in mathematics (Princeton, NJ, 16–18 March 1970). Annals of Mathematics Studies 70. Princeton University Press, 1971. A Russian translation was published in Matematika 18:1 (1974). MR 343319 Zbl 0247.58011 incollection

F. Hirzebruch	Related	Volume
Lars Hörmander	Related
John W. Milnor	Related
Jean-Pierre Serre	Related

[41] L. A. Coburn, R. G. Douglas, D. G. Schaeffer, and I. M. Singer: “\( C^* \)-algebras of operators on a half-space, II: Index theory,” Inst. Hautes Études Sci. Publ. Math. 40 (1971), pp. 69–79. MR 358418 Zbl 0241.47027 article

Lewis Alan Coburn	Related
David George Schaeffer	Related
Ronald George Douglas	Related

[42] I. M. Singer: “Operator theory and periodicity,” pp. 949–951 in Proceedings of an international symposium on operator theory (Bloomington, IN, 1–5 June 1970), published as Indiana Univ. Math. J. 20 : 10. Issue edited by P. R. Halmos, E. Hopf, and G. Springer. April 1971. MR 445304 Zbl 0228.55020 incollection

Paul Richard Halmos	Related
Eberhard Friedrich Ferdinand Hopf	Related
George Springer	Related

[43] M. F. Atiyah and I. M. Singer: “The index of elliptic operators, V,” Ann. Math. (2) 93 : 1 (January 1971), pp. 139–149. A Russian translation was published in Uspehi Mat. Nauk 27:4(166) (1972). MR 279834 article

[44] L. A. Coburn, R. G. Douglas, and I. M. Singer: “An index theorem for Wiener–Hopf operators on the discrete quarter-plane,” J. Diff. Geom. 6 : 4 (1972), pp. 587–593. This issue is dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays. MR 321142 Zbl 0252.47026 article

S. S. Chern	Related	Volume
Ronald George Douglas	Related
Lewis Alan Coburn	Related
Donald Clayton Spencer	Related

[45] M. F. At’ja and I. M. Zinger: “The index of elliptic operators, IV,” Uspehi Mat. Nauk 27 : 4(166) (1972), pp. 161–178. Russian translation of an article published in Ann. Math. (2) 93:1 (1971). MR 385933 Zbl 0237.58017 article

[46] M. F. Atiyah and I. M. Singer: “The index of elliptic operators, V,” Usp. Mat. Nauk 27 : 4(166) (1972). Russian translation of an article published in Ann. Math. (2) 93:1 (1971). Zbl 0237.58018 article

[47] M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry,” Bull. London Math. Soc. 5 (July 1973), pp. 229–234. MR 331443 Zbl 0268.58010 article

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

[48] D. B. Ray and I. M. Singer: “Analytic torsion,” pp. 167–181 in Partial differential equations (Berkeley, CA, 9–27 August 1971). Edited by D. C. Spencer. Proceedings of Symposia in Pure Mathematics 23. American Mathematical Society (Providence, RI), 1973. MR 339293 Zbl 0273.58014 incollection

Daniel Burrill Ray	Related
Donald Clayton Spencer	Related

[49] I. M. Singer: “Recent applications of index theory for elliptic operators,” pp. 11–31 in Partial differential equations (Berkeley, CA, 9–27 August 1971). Edited by D. C. Spencer. Proceedings of Symposia in Pure Mathematics 23. American Mathematical Society (Providence, RI), 1973. MR 341538 Zbl 0293.58013 incollection

[50] D. B. Ray and I. M. Singer: “Analytic torsion for complex manifolds,” Ann. Math. (2) 98 (July 1973), pp. 154–177. MR 383463 Zbl 0267.32014 article

In [Ray and Singer 1973] we investigated the analytic analogue of Reidemeister–Franz torsion for a compact \( C^{\infty} \) manifold \( W \). When \( W \) is without boundary it is an invariant associated to the de Rham complex with coefficients in a flat vector bundle. In this paper, we study the corresponding invariants, \( \overline{\partial} \)-torsion, for the \( \overline{\partial} \)-complex of a complex analytic manifold. The definition, given in §1, depends on a choice of Hermitian metric. In §2, among other things, we prove that the ratios of the \( \overline{\partial} \)-torsions are independent of the metric when the complexes are acyclic (of course they do depend on the complex structure). The formal properties of \( \overline{\partial} \)-torsion are similar to those of ordinary \( R \)-torsion and are described in §3. For Riemann surfaces, our invariant can be expressed in terms of functions of some interest in number theory: for genus 1 in terms of the elliptic theta functions and for surfaces of higher genus in terms of Selberg’s zeta function. These matters are discussed in §4. We conclude with a few examples of dimension greater than one.

[51] L. A. Coburn, R. D. Moyer, and I. M. Singer: “\( C^* \)-algebras of almost periodic pseudo-differential operators,” Acta Math. 130 (1973), pp. 279–307. MR 415407 Zbl 0263.47042 article

Lewis Alan Coburn	Related
Robert Dale Moyer	Related

[52] I. M. Singer: “Future extensions of index theory and elliptic operators,” Matematika 18 : 1 (1974), pp. 52–61. Russian translation of an article published in Prospects in mathematics (1971). Zbl 0286.58011 article

[53] M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, I,” Math. Proc. Cambridge Philos. Soc. 77 (1975), pp. 43–69. MR 397797 Zbl 0297.58008 article

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

[54] M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, II,” Math. Proc. Cambridge Philos. Soc. 78 : 3 (November 1975), pp. 405–432. MR 397798 Zbl 0314.58016 article

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

[55] I. M. Singer: “Eigenvalues of the Laplacian and invariants of manifolds,” pp. 187–200 in Proceedings of the International Congress of Mathematicians (Vancouver, BC, 21–29 August 1974), vol. 1. Edited by R. D. James. Canadian Mathematical Congress (Montreal), 1975. MR 420746 Zbl 0345.58014 incollection

[56] J. J. Duistermaat and I. M. Singer: “Isomorphismes entre algèbres d’opérateurs pseudo-différentiels” [Isomorphisms between algebras of pseudo-differential operators] in Équations aux dérivées partielles linéaires et non linéaires [Linear and nonlinear partial differential equations]. École Polytechnique (Paris), 1975. Goulaouic–Lions–Schwartz seminar 1974–1975, Exposé no. 24, 8 pp. MR 482869 Zbl 0306.58017 incollection

[57] M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, III,” Math. Proc. Cambridge Philos. Soc. 79 : 1 (1976), pp. 71–99. MR 397799 Zbl 0325.58015 article

In Parts I and II of this paper [1975a; 1975b] we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator \( A \) on a compact manifold we defined \[ \eta_A(s) = \sum_{\lambda\neq 0}\operatorname{sign} \lambda \,|\lambda|^{-s}, \] where \( \lambda \) runs over the eigenvalues of \( A \). For the particular operators of interest in Riemannian geometry we showed that \( \eta_A(s) \) had an analytic continuation to the whole complex \( s \)-plane, with simple poles, and that \( s = 0 \) was not a pole. The real number \( \eta_A(0) \), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

[58] J. J. Duistermaat and I. M. Singer: “Order-preserving isomorphisms between algebras of pseudo-differential operators,” Comm. Pure Appl. Math. 29 : 1 (1976), pp. 39–47. MR 402830 Zbl 0317.58017 article

[59] I. M. Singer and J. A. Thorpe: Lecture notes on elementary topology and geometry, reprint edition. Undergraduate Texts in Mathematics. Springer (New York), 1976. Reprint of 1967 original. MR 413152 Zbl 0342.54002 book

[60] \( K \)-theory and operator algebras (Athens, GA, 21–25 April 1975). Edited by B. B. Morrel and I. M. Singer. Lecture Notes in Mathematics 575. Springer (Berlin), 1977. MR 457035 Zbl 0337.00005 book

[61] M. F. Atiyah, N. J. Hitchin, and I. M. Singer: “Deformations of instantons,” Proc. Nat. Acad. Sci. U.S.A. 74 : 7 (July 1977), pp. 2662–2663. MR 458424 Zbl 0356.58011 article

Nigel J. Hitchin	Related
M. F. Atiyah	Related	Volume

[62] I. M. Singer: “Some remarks on operator theory and index theory,” pp. 128–138 in \( K \)-theory and operator algebras (Athens, GA, 21–25 April 1975). Edited by B. B. Morrel and I. M. Singer. Lecture Notes in Mathematics 575. Springer (Berlin), 1977. MR 467848 Zbl 0444.47040 incollection

[63] I. M. Singer: “Some remarks on the Gribov ambiguity,” Comm. Math. Phys. 60 : 1 (February 1978), pp. 7–12. MR 500248 Zbl 0379.53009 article

[64] M. F. Atiyah, N. J. Hitchin, and I. M. Singer: “Self-duality in four-dimensional Riemannian geometry,” Proc. Roy. Soc. London Ser. A 362 : 1711 (September 1978), pp. 425–461. MR 506229 Zbl 0389.53011 article

Nigel J. Hitchin	Related
M. F. Atiyah	Related	Volume

[65] G. B. Kolata: “Isadore Singer and differential geometry,” Science 204 : 4396 (1 June 1979), pp. 933–934. MR 531490 Zbl 1225.01074 article

[66] 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 609553 Zbl 0443.00012 book

Joseph Albert Wolf	Related
S. S. Chern	Related	Volume
Shōshichi Kobayashi	Related	Volume
Alan Weinstein	Related
Wu-Chung Hsiang	Related
Hung-hsi Wu	Related

[67] I. M. Singer: “The geometry of the orbit space for nonabelian gauge theories,” Phys. Scr. 24 : 5 (1981), pp. 817–820. MR 639408 Zbl 1063.81623 article

[68] M. F. Atiyah, H. Donnelly, and I. M. Singer: “Geometry and analysis of Shimizu \( L \)-functions,” Proc. Nat. Acad. Sci. U.S.A. 79 : 18 (September 1982), pp. 5751. MR 674920 Zbl 0503.12007 article

M. F. Atiyah	Related	Volume
Harold Gerard Donnelly	Related

[69] I. M. Singer: “On Yang–Mills fields,” pp. 35–50 in Nonlinear problems: Present and future (Los Alamos, NM, 2–6 March 1981). Edited by A. Bishop, D. Campbell, and B. Nicolaenko. North-Holland Mathematics Studies 61. North-Holland (Amsterdam), 1982. MR 675626 Zbl 0499.35092 incollection

Alan Bishop	Related
David Kelly Campbell	Related
Basil Nicolaenko	Related

[70] Banach space theory and its applications: Proceedings of the first Romanian–GDR seminar (Bucharest, Romania, 31 August–6 September 1981). Edited by A. Pietsch, N. Popa, and I. Singer. Lecture Notes in Mathematics 991. Springer (Berlin), 1983. MR 714167 Zbl 0504.00015 book

Nicolae Popa	Related
Albrecht Pietsch	Related

[71] M. F. Atiyah, H. Donnelly, and I. M. Singer: “Eta invariants, signature defects of cusps, and values of \( L \)-functions,” Ann. Math. (2) 118 : 1 (July 1983), pp. 131–177. An addendum to this article was published in Ann. Math. (2) 119:3 (1984). MR 707164 Zbl 0531.58048 article

M. F. Atiyah	Related	Volume
Harold Gerard Donnelly	Related

[72] M. F. Atiyah and I. M. Singer: “Dirac operators coupled to vector potentials,” Proc. Nat. Acad. Sci. U.S.A. 81 : 8 (April 1984), pp. 2597–2600. MR 742394 Zbl 0547.58033 article

[73] M. F. Atiyah, H. Donnelly, and I. M. Singer: “Signature defects of cusps and values of \( L \)-functions: The nonsplit case,” Ann. Math. (2) 119 : 3 (1984), pp. 635–637. Addendum to an article published in Ann. Math. (2) 118:1 (1983). MR 744866 Zbl 0577.58030 article

M. F. Atiyah	Related	Volume
Harold Gerard Donnelly	Related

[74] O. Alvarez, I. M. Singer, and B. Zumino: “Gravitational anomalies and the family’s index theorem,” Comm. Math. Phys. 96 : 3 (1984), pp. 409–417. MR 769356 Zbl 0587.58042 article

Bruno Zumino	Related
Orlando Alvarez	Related

[75] Vertex operators in mathematics and physics (Berkeley, CA, 10–17 November 1983). Edited by J. Lepowsky, S. Mandelstam, and I. M. Singer. Mathematical Sciences Research Institute Publications 3. Springer (New York), 1985. MR 781369 Zbl 0549.00013 book

James Ivan Lepowsky	Related
Stanley Mandelstam	Related

[76] I. M. Singer, B. Wong, S.-T. Yau, and S. S.-T. Yau: “An estimate of the gap of the first two eigenvalues in the Schrödinger operator,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 : 2 (1985), pp. 319–333. MR 829055 Zbl 0603.35070 article

Stephen Shing-Toung Yau	Related
Bun Wong	Related
Shing-Tung Yau	Related

[77] I. M. Singer: “Families of Dirac operators with applications to physics,” pp. 323–340 in Élie Cartan et les mathématiques d’aujourd’hui: The mathematical heritage of Élie Cartan [Élie Cartan and the mathematics of today: The mathematical heritage of Élie Cartan] (Lyon, France, 25–29 June 1984). Astérisque. Société Mathématique de France (Paris), 1985. MR 837207 Zbl 0603.58054 incollection

[78] I. M. Singer: “Differential geometry, fiber bundles and physical theories,” Phys. Today 35 : 3 (1985), pp. 41–44. A Bulgarian translation was published in Fiz.-Mat. Spis. 27 (1985). article

Among intellectual disciplines, mathematics occupies a unique position. It is in many respects an art, but it is also the language of science. Although a great deal of mathematics can be traced directly to external influences, much of its creativity is motivated internally: “pour la gloire de l’esprit humain,” as Carl G. T. Jacobi put it. Often the mathematician lets the imagination soar, constrained only by logic, intrinsic structure, and a sense of historical continuity. Yet from time to time these abstract deliberations have important applications in other fields. Working on purely abstract problems in geometry, mathematicians have independently found a suitable framework for the gauge theories that appear to describe elementary particles.

[79] I. M. Singer: “Differential geometry, fiber bundles and physical theories,” Fiz.-Mat. Spis. 27 (1985), pp. 136–145. Bulgarian translation of an article published in Phys. Today 35:3 (1985). Zbl 0661.53058 article

[80] G. Moore, J. Harris, P. Nelson, and I. Singer: “Modular forms and the cosmological constant,” Phys. Lett. B 178 : 2–3 (October 1986), pp. 167–173. MR 858672 article

Gregory W. Moore	Related
Philip Charles Nelson	Related
Joseph Daniel Harris	Related

[81] I. M. Singer: “The \( \eta \)-invariant and the index,” pp. 239–258 in Mathematical aspects of string theory (San Diego, CA, 21 July–1 August 1986). Edited by S.-T. Yau. Advanced Series in Mathematical Physics 1. World Scientific (Singapore), 1987. MR 915824 Zbl 0669.58036 incollection

[82] L. Baulieu and I. M. Singer: “Topological Yang–Mills symmetry,” pp. 12–19 in Conformal field theories and related topics (Annecy-le-Vieux, France, 14–16 March 1988), published as Nuclear Phys. B Proc. Suppl. 5B : 2. Issue edited by R. Stora, P. Binétruy, and P. Sorba. North-Holland (Amsterdam), December 1988. MR 1002954 Zbl 0958.58500 incollection

Paul Sorba	Related
Laurent Baulieu	Related
Pierre Binétruy	Related
Raymond Stora	Related

[83] R. Seeley and I. M. Singer: “Extending \( \overline\partial \) to singular Riemann surfaces,” J. Geom. Phys. 5 : 1 (1988), pp. 121–136. MR 1027535 Zbl 0692.30038 article

[84] G. Moore, J. Harris, P. Nelson, and I. Singer: “Errata: ‘Modular forms and the cosmological constant’,” Phys. Lett. B 201 : 4 (February 1988), pp. 579. Errata to an article published in Phys. Lett. B 178:2–3 (1986). MR 929610 article

Gregory W. Moore	Related
Philip Charles Nelson	Related
Joseph Daniel Harris	Related

[85] I. M. Singer: “Some problems in the quantization of gauge theories and string theories,” pp. 199–216 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. MR 974336 Zbl 0702.58010 incollection

Hermann Weyl	Related
Raymond O'Neil Wells, Jr	Related

[86] L. Baulieu and I. M. Singer: “The topological sigma model,” Comm. Math. Phys. 125 : 2 (June 1989), pp. 227–237. MR 1016870 Zbl 0684.57016 article

[87] T. R. Ramadas, I. M. Singer, and J. Weitsman: “Some comments on Chern–Simons gauge theory,” Comm. Math. Phys. 126 : 2 (December 1989), pp. 409–420. MR 1027504 Zbl 0686.53066 article

T. R. Ramadas	Related
Jonathan Weitsman	Related

[88] M. Andler: “Entretien avec Isadore Singer” [Interview with Isadore Singer], Gaz. Math. 43 (1990), pp. 3–10. MR 1035386 article

[89] O. Alvarez, I. M. Singer, and P. Windey: “Quantum mechanics and the geometry of the Weyl character formula,” Nuclear Phys. B 337 : 2 (June 1990), pp. 467–486. MR 1057724 article

General field theoretic methods are developed which will allow a path integral derivation of the character formula for loop groups. The methods are introduced in the classical Weyl character case. The irreducible representations of a compact semi-simple Lie group \( G \) are realized as the ground states of a supersymmetric quantum mechanical system. The Hilbert space for the quantum mechanical system is the space of sections of a holomorphic line bundle \( L \) over the complex manifold \( G/T \), where \( T \) is the maximal torus of \( G \). The Weyl character formula is derived by an explicit path integral computation of the index of the Dolbeault operator.

Orlando Alvarez	Related
Paul Windey	Related

[90] The legacy of John von Neumann (Hempstead, NY, 29 May–4 June 1988). Edited by J. Glimm, J. Impagliazzo, and I. Singer. Proceedings of Symposia in Pure Mathematics 50. American Mathematical Society (Providence, RI), 1990. Zbl 0699.00010 book

James G. Glimm	Related
John von Neumann	Related
John Impagliazzo	Related

[91] L. Baulieu and I. M. Singer: “Conformally invariant gauge fixed actions for 2-D topological gravity,” Comm. Math. Phys. 135 : 2 (1991), pp. 253–265. MR 1087384 Zbl 0725.53071 article

[92] I. M. Singer: “The new mathematical physics,” pp. 187–191 in Elementary particles and the universe: Essays in honor of Murray Gell-Mann (Pasadena, CA, 27–28 September 1989). Edited by J. H. Schwarz. Cambridge University Press, 1991. MR 1145030 incollection

Murray Gell-Mann	Related
John Henry Schwarz	Related

[93] Geometry and physics: Essays in honour of I. M. Gelfand on the occasion of his 75th birthday. Edited by S. Gindikin and I. M. Singer. North-Holland and Pitagora Editrice (Amsterdam and Bologna), 1991. Reprinted from the Journal of Geometry and Physics, 5 (1988). Zbl 0745.00067 book

Simon Grigorevich Gindikin	Related
Israïl Moiseevich Gelfand	Related

[94] O. Alvarez, I. M. Singer, and P. Windey: “The supersymmetric \( \sigma \)-model and the geometry of the Weyl–Kač character formula,” Nuclear Phys. B 373 : 3 (April 1992), pp. 647–687. MR 1159678 ArXiv hep-th/9109047 article

Orlando Alvarez	Related
Paul Windey	Related

[95] I. M. Singer: “S. S. Chern at Chicago,” pp. 88–90 in Chern: A great geometer of the twentieth century. Edited by S.-T. Yau. International Press (Hong Kong), 1992. MR 1201344 incollection

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

[96] S. Axelrod and I. M. Singer: “Chern–Simons perturbation theory,” pp. 3–45 in Proceedings of the XXth international conference on differential geometric methods in theoretical physics (New York, 3–7 June 1991), vol. 1. Edited by S. Catto and A. Rocha. World Scientific (River Edge, NJ), 1992. MR 1225107 Zbl 0813.53051 ArXiv hep-th/9110056 incollection

We study the perturbation theory for three dimensional Chern–Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the action obtained by BRS gauge fixing in the Lorentz gauge has a superspace formulation. The basic properties of the propagator and the Feynman rules are written in a precise manner in the language of differential forms. Using the explicit description of the propagator singularities, we prove that the theory is finite. Finally the anomalous metric dependence of the 2-loop partition function on the Riemannian metric (which was introduced to define the gauge fixing) can be cancelled by a local counterterm as in the 1-loop case. In fact, the counterterm is equal to the Chern–Simons action of the metric connection, normalized precisely as one would expect based on the framing dependence of Witten’s exact solution.

Scott Elliot Axelrod	Related
Alvany Rocha-Caridi	Related
Sultan Catto	Related

[97] S. Axelrod and I. M. Singer: “Chern–Simons perturbation theory, II,” J. Diff. Geom. 39 : 1 (1994), pp. 173–213. Also published in Perspectives in mathematical physics (1994). Part I was published in Proceedings of the XXth international conference on differential geometric methods in theoretical physics (1992). MR 1258919 Zbl 0827.53057 ArXiv hep-th/9304087 article

[98] I. M. Singer: The current interface of geometry and elementary particle physics, 1994. 90 minute videocassette. A joint AMS-MAA lecture presented in Baltimore, Maryland, 11 January 1992. Introduced by Franklin P. Peterson. MR 1284042 Zbl 0803.53002 misc

[99]S. Axelrod and I. M. Singer: “Chern–Simons perturbation theory. II,” pp. 17–49 in Perspectives in mathematical physics. Conf. Proc. Lecture Notes Math. Phys. III. Int. Press (Cambridge, MA), 1994. MR 1314660 Zbl 0889.53053

[100] I. M. Singer: “On the master field in two dimensions,” pp. 263–281 in Functional analysis on the eve of the 21st century: In honor of the eightieth birthday of I. M. Gelfand (New Brunswick, NJ, 24–27 October 1993), vol. 1. Edited by S. Gindikin, J. Lepowsky, and R. L. Wilson. Progress in Mathematics 131. Birkhäuser (Boston), 1995. MR 1373007 Zbl 0861.53075 incollection

Israïl Moiseevich Gelfand	Related
Simon Grigorevich Gindikin	Related
James Ivan Lepowsky	Related
Robert Lee Wilson	Related

[101] 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
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

[102] I. M. Singer and H. Wu: “A tribute to Warren Ambrose,” Notices Amer. Math. Soc. 43 : 4 (April 1996), pp. 425–427. MR 1379775 Zbl 1044.01546 article

Hung-hsi Wu	Related
Warren Arthur Ambrose	Related

[103] I. M. Singer: “Tribute to I. M. Gelfand for his 80th birthday celebration,” pp. xix–xxii in Functional analysis on the eve of the 21st century: In honor of the eightieth birthday of I. M. Gelfand (New Brunswick, NJ, 24–27 October 1993), vol. 1. Edited by S. Gindikin, J. Lepowsky, and R. L. Wilson. Progress in Mathematics 132. Birkhäuser (Boston), 1996. MR 1389018 Zbl 0841.01019 incollection

Israïl Moiseevich Gelfand	Related
Simon Grigorevich Gindikin	Related
James Ivan Lepowsky	Related
Robert Lee Wilson	Related

[104] The legacy of Norbert Wiener: A centennial symposium (Cambridge, MA, 8–14 October 1994). Edited by D. Jerison, I. M. Singer, and D. W. Stroock. Proceedings of Symposia in Pure Mathematics 60. American Mathematical Society (Providence, RI), 1997. MR 1460268 Zbl 0869.00026 book

Daniel W. Stroock	Related
Norbert Wiener	Related
David Saul Jerison	Related

[105] 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
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related
David Saul Jerison	Related

[106] Current developments in mathematics 1997 (Cambridge, MA, 1997), preliminary edition. Edited by R. Bott, A. Jaffe, S.-T. Yau, D. Jerison, and I. Singer. International Press (Cambridge, MA), 1997. Preliminary version of a book eventually published in 1999. Zbl 0899.00016 book

Arthur Jaffe	Related
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related
David Saul Jerison	Related

[107] L. Baulieu, H. Kanno, and I. M. Singer: “Special quantum field theories in eight and other dimensions,” Comm. Math. Phys. 194 : 1 (May 1998), pp. 149–175. MR 1628298 Zbl 0910.53054 article

We build nearly topological quantum field theories in various dimensions. We give special attention to the case of eight dimensions for which we first consider theories depending only on Yang–Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in \( SO(8) \), namely \( SU(4) \) and \( Spin(7) \). The choice of \( SU(4) \) gives a quantum field theory for a Calabi–Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of \( Spin(7) \) defines another eight dimensional theory for a Joyce manifold which could be of relevance in \( M \)- and \( F \)-theories. Relations to the eight dimensional supersymmetric Yang–Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg–Witten equations. The latter are thus related to pure Yang–Mills self-duality equations in 8 dimensions as well as to the \( N=1 \), \( D=10 \) super Yang–Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions.

Hiroaki Kanno	Related
Laurent Baulieu	Related

[108] L. Baulieu, H. Kanno, and I. M. Singer: “Cohomological Yang–Mills theory in eight dimensions,” pp. 365–373 in Dualities in gauge and string theories (Seoul and Sokcho, S. Korea, 17–28 February 1997). Edited by Y. M. Cho and S. Nam. World Scientific (River Edge, NJ), 1998. MR 1712267 Zbl 1052.53508 incollection

We construct nearly topological Yang–Mills theories on eight dimensional manifolds with a special holonomy group. These manifolds are the Joyce manifold with \( \operatorname{Spin}(7) \) holonomy and the Calabi–Yau manifold with \( \operatorname{SU}(4) \) holonomy. An invariant closed four form \( T_{\mu\nu\rho\sigma} \) on the manifold allows us to define an analogue of the instanton equation, which serves as a topological gauge fixing condition in BRST formalism. The model on the Joyce manifold is related to the eight dimensional supersymmetric Yang–Mills theory. Topological dimensional reduction to four dimensions gives non-abelian Seiberg–Witten equation.

Laurent Baulieu	Related
Hiroaki Kanno	Related
Y. M. Cho	Related
Soonkeon Nam	Related

[109] 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
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related
David Saul Jerison	Related

[110] Current developments in mathematics 1998 (Cambridge, MA, 1998). Edited by B. Mazur, W. Schmid, S.-T. Yau, D. Jerison, I. M. Singer, and D. Stroock. Current Developments in Mathematics. International Press (Somerville, MA), 1999. MR 1772320 Zbl 0963.00022 book

Daniel W. Stroock	Related
Barry C. Mazur	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[111] Current developments in mathematics 1999 (Cambridge, MA, 1999). Edited by B. Mazur, W. Schmid, S.-T. Yau, D. Jerison, I. M. Singer, and D. Stroock. Current Developments in Mathematics. International Press (Somerville, MA), 1999. MR 1990245 Zbl 1051.00008 book

Daniel W. Stroock	Related
Barry C. Mazur	Related
Wilfried Schmid	Related
Shing-Tung Yau	Related
David Saul Jerison	Related

[112] “2000 Steele Prizes,” Notices Amer. Math. Soc. 47 : 4 (April 2000), pp. 477–480. MR 1745621 Zbl 1040.01503 article

John Horton Conway	Related
Barry C. Mazur	Related

[113] R. V. Kadison: “Which Singer is that?,” pp. 347–373 in Papers dedicated to Atiyah, Bott, Hirzebruch and Singer (Cambridge, MA, 14–16 May 1999). Surveys in Differential Geometry 7. International Press (Somerville, MA), 2000. Reprinted in The founders of index theory (2009). MR 1919431 Zbl 1075.01010 incollection

F. Hirzebruch	Related	Volume
M. F. Atiyah	Related	Volume
Richard V. Kadison	Related
Raoul H. Bott	Related	Volume

[114] Papers dedicated to Atiyah, Bott, Hirzebruch and Singer (Cambridge, MA, 14–16 May 1999). Surveys in Differential Geometry 7. International Press (Somerville, MA), 2000. Zbl 1044.53002 book

Raoul H. Bott	Related	Volume
M. F. Atiyah	Related	Volume
F. Hirzebruch	Related	Volume

[115] O. Alvarez and I. M. Singer: “Beyond the elliptic genus,” Nuclear Phys. B 633 : 3 (July 2002), pp. 309–344. MR 1910266 Zbl 0995.58016 article

[116] 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
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

[117] “Atiyah and Singer receive 2004 Abel Prize,” Notices Amer. Math. Soc. 51 : 6 (2004), pp. 649–650. MR 2064152 Zbl 1168.01308 article

[118] M. Raussen and C. Skau: “Interview with Michael Atiyah and Isadore Singer,” Mitt. Dtsch. Math.-Ver. 12 : 4 (2004), pp. 272–281. Originally published in the European Mathematical Society Newsletter, September 2004. Also published in Notices Amer. Math. Soc. 52:2 (2005). MR 2110686 Zbl 1087.01013 article

M. F. Atiyah	Related	Volume
Martin Raussen	Related
Christian Skau	Related

[119] M. Raussen and C. Skau: “Interview with Michael Atiyah and Isadore Singer,” Eur. Math. Soc. Newsl. 53 (September 2004), pp. 24–30. The interview took place in Oslo, on the 24th of May 2004, prior to the Abel prize celebrations. Also published in Notices Amer. Math. Soc. 52:2 (2005) and Mitt. Dtsch. Math.-Ver. 12:4 (2004). article

M. F. Atiyah	Related	Volume
Martin Raussen	Related
Christian Skau	Related

[120] “Honorary member 2004: Isadore M. Singer,” Bull. London Math. Soc. 37 : 3 (June 2005), pp. 467. article

[121] M. Raussen and C. Skau: “Interview with Michael Atiyah and Isadore Singer,” Notices Amer. Math. Soc. 52 : 2 (February 2005), pp. 225–233. Originally published in the European Mathematical Society Newsletter, September 2004. Also published in Mitt. Dtsch. Math.-Ver. 12:4 (2004). MR 2110633 Zbl 1142.01347 article

M. F. Atiyah	Related	Volume
Martin Raussen	Related
Christian Skau	Related

[122] V. Mathai, R. B. Melrose, and I. M. Singer: “The index of projective families of elliptic operators,” Geom. Topol. 9 : 1 (2005), pp. 341–373. MR 2140985 Zbl 1083.58021 ArXiv math.DG/0206002 article

Varghese Mathai	Related
Richard Burt Melrose	Related

[123] M. J. Hopkins and I. M. Singer: “Quadratic functions in geometry, topology, and M-theory,” J. Diff. Geom. 70 : 3 (2005), pp. 329–452. MR 2192936 Zbl 1116.58018 ArXiv math/0211216 article

[124] The unity of mathematics: In honor of the ninetieth birthday of I. M. Gelfand (Cambridge, MA, 31 August–4 September 2003). Edited by P. Etingof, V. Retakh, and I. M. Singer. Progress in Mathematics 244. Birkhäuser (Boston, MA), 2006. MR 2182597 Zbl 1083.00015 book

Vladimir Solomonovich Retakh	Related
Pavel Ilyich Ètingof	Related
Israïl Moiseevich Gelfand	Related

[125] V. Mathai, R. B. Melrose, and I. M. Singer: “Fractional analytic index,” J. Diff. Geom. 74 : 2 (2006), pp. 265–292. MR 2258800 Zbl 1115.58021 ArXiv math/0402329 article

For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier–Douady, 3-class, \( w \), on the manifold, we define the ring of differential operators ‘acting on sections of the projective bundle’ in a formal sense. In particular, any oriented even-dimensional manifold carries a projective spin Dirac operator in this sense. More generally the corresponding space of pseudodifferential operators is defined, with supports sufficiently close to the diagonal, i.e., the identity relation. For such elliptic operators we define the numerical index in an essentially analytic way, as the trace of the commutator of the operator and a parametrix and show that this is homotopy invariant. Using the heat kernel method for the twisted, projective spin Dirac operator, we show that this index is given by the usual formula, now in terms of the twisted Chern character of the symbol, which in this case defines an element of \( K \)-theory twisted by \( w \); hence the index is a rational number but in general it is not an integer.

Varghese Mathai	Related
Richard Burt Melrose	Related

[126] Quantum field theory, supersymmetry, and enumerative geometry. Edited by D. S. Freed, D. R. Morrison, and I. Singer. IAS/Park City Mathematics Series 11. American Mathematical Society (Providence, RI), 2006. MR 2265374 Zbl 1122.81010 book

[127] V. Mathai, R. B. Melrose, and I. M. Singer: “Equivariant and fractional index of projective elliptic operators,” J. Diff. Geom. 78 : 3 (2008), pp. 465–473. MR 2396250 Zbl 1147.58018 ArXiv math/0611819 article

Varghese Mathai	Related
Richard Burt Melrose	Related

[128] D. Freed and J. Lott: “Singer’s Berkeley seminar,” pp. 277–278 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. incollection

F. Hirzebruch	Related	Volume
John William Lott	Related
M. F. Atiyah	Related	Volume
Daniel Stuart Freed	Related
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

[129] K. Hoffman: “Fifty years of friendship and collaboration with Singer,” pp. 269–272 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. incollection

F. Hirzebruch	Related	Volume
Kenneth Hoffman	Related
M. F. Atiyah	Related	Volume
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

[130] R. V. Kadison: “Which Singer is that?,” pp. 279–302 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. Originally published in Surveys in Differential Geometry 7 (2000). incollection

F. Hirzebruch	Related	Volume
M. F. Atiyah	Related	Volume
Richard V. Kadison	Related
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

[131] 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

F. Hirzebruch	Related	Volume
M. F. Atiyah	Related	Volume
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

[132] V. Mathai, R. B. Melrose, and I. M. Singer: “The index of projective families of elliptic operators: The decomposable case,” pp. 255–296 in From probability to geometry, II: Volume in honor of the 60th birthday of Jean-Michel Bismut. Edited by X. Dai, R. Léandre, X. Ma, and W. Zhang. Astérisque 328. Société Mathématique de France (Paris), 2009. MR 2674880 Zbl 1207.19006 ArXiv 0809.0028 incollection

An index theory for projective families of elliptic pseudodifferential operators is developed under two conditions. First, that the twisting, i.e. Dixmier–Douady, class is in \[ H^2(X;\mathbb{Z}) \cup H^1(X;\mathbb{Z}) \subset H^3(X;\mathbb{Z}) \] and secondly that the 2-class part is trivialized on the total space of the fibration. One of the features of this special case is that the corresponding Azumaya bundle can be refined to a bundle of smoothing operators. The topological and the analytic ndex of a projective family of elliptic operators associated with the smooth Azumaya bundle both take values in twisted \( K \)-theory of the parameterizing space and the main result is the equality of these two notions of index. The twisted Chern character of the index class is then computed by a variant of Chern–Weil theory.

Xianzhe Dai	Related
Xiaonan Ma	Related
Varghese Mathai	Related
Richard Burt Melrose	Related
Rémi Léandre	Related
Weiping Zhang	Related
Jean-Michel Bismut	Related

[133] C. C. Moore: “Iz Singer and the founding of MSRI,” pp. 273–276 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. incollection

F. Hirzebruch	Related	Volume
M. F. Atiyah	Related	Volume
Calvin C. Moore	Related	Volume
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

[134] H. Rossi: “Reminiscences of a mathematics student, 40 years ago,” pp. 303–306 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. incollection

F. Hirzebruch	Related	Volume
M. F. Atiyah	Related	Volume
Raoul H. Bott	Related	Volume
Hugo Rossi	Related
Shing-Tung Yau	Related

[135] Isadore Singer, August 2009. Online video interview, Simons Foundation “Science Lives” series. misc

[136] D. Stroock: “Is Singer,” pp. 307–310 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. incollection

F. Hirzebruch	Related	Volume
M. F. Atiyah	Related	Volume
Daniel W. Stroock	Related
Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

[137] F. Warner: “I. M. Singer,” pp. 311–312 in The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer. Edited by S.-T. Yau. International Press (Somerville, MA), 2009. incollection

F. Hirzebruch	Related	Volume
M. F. Atiyah	Related	Volume
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[141] Perspectives in mathematics and physics: Essays dedicated to Isadore Singer’s 85th birthday. Edited by T. Mrowka and S.-T. Yau. Surveys in Differential Geometry 15. International Press (Somerville, MA), 2011. MR 2841268 Zbl 1226.00040 book

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[142] S.-T. Yau, F. E. P. Hirzebruch, M. Atiyah, M. do Carmo, R. E. Greene, W.-l. Huang, K. Reich, L. Ji, J. Li, E. Kuh, Y. R. Shen, I. M. Singer, A. Weinstein, and J. A. Wolf: “Shiing-Shen Chern (1911–2004),” Notices Am. Math. Soc. 58 : 9 (October 2011), pp. 1226–1249. Hirzebruch’s contribution to this article was also published in Frontiers of mathematical sciences (2011). MR 2868099 Zbl 1228.01052 article

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@article {key2868099m,
	AUTHOR = {Yau, Shing-Tung and Hirzebruch, Friedrich
		 Ernst Peter and Atiyah, Michael and
		 do Carmo, Manfredo and Greene, Robert
		 E. and Huang, Wen-ling and Reich, Karin
		 and Ji, Lizhen and Li, Jun and Kuh,
		 Ernest and Shen, Y. Ron and Singer,
		 Isadore M. and Weinstein, Alan and Wolf,
		 Joseph A.},
	TITLE = {Shiing-{S}hen {C}hern (1911--2004)},
	JOURNAL = {Notices Am. Math. Soc.},
	FJOURNAL = {Notices of the American Mathematical
		 Society},
	VOLUME = {58},
	NUMBER = {9},
	MONTH = {October},
	YEAR = {2011},
	PAGES = {1226--1249},
	URL = {http://www.ams.org/notices/201109/rtx110901226p.pdf},
	NOTE = {Hirzebruch's contribution to this article
		 was also published in \textit{Frontiers
		 of mathematical sciences} (2011). MR:2868099.
		 Zbl:1228.01052.},
	ISSN = {0002-9920},
}

[143] I. M. Singer, D. Kazhdan, A. Vershik, B. Kostant, S. Gindikin, P. Lax, and A. Zelevinsky: “Israel Moiseevich Gelfand, I.” Edited by V. Retakh. Notices Amer. Math. Soc. 60 : 1 (2013), pp. 24–49. MR 3052462 Zbl 1290.01027 article

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