Skip to main content
SpringerLink
Log in

Index Theorems

  • Chapter
Partial Differential Equations VIII

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 65))

Abstract

The theory of the index of elliptic operators has for a long time been developed in parallel within the framework of two branches of mathematics that, traditionally, are regarded as quite far apart. One of them is the the theory of elliptic equations and boundary value problems—in particular, the theory of singular integral equations. The other is topology and algebraic geometry, where very specific elliptic operators have been considered. A significant role in bringing these two domains together was played by Gel’fand (1960), who posed the problem of topological classification of elliptic operators, in particular, the computation of the index in topological terms. The latter was fully solved by Atiyah and Singer in 1963. The Atiyah-Singer theorem has generated a tremendous amount of interest, which has continued to this day and has exercised an immense influence on the subsequent development and convergence of the theory of differential equations and topology. Thus, for example, the necessity to extend the class of deformations of elliptic operators has led to new algebras of pseudodifferential operators (PDOs). In topology, the Atiyah-Singer theorem has stimulated the further development of K-theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alvarez-Gaumé, L. (1984): Supersymmetry and the Atiyah-Singer index theorem. Physica A 124, 29–45. Zbl. 599. 58042

    Google Scholar 

  • Atiyah, M.F. (1967): K-Theory. Benjamin, New York. Zbl. 159, 533

    Google Scholar 

  • Atiyah, M.F. (1974): Elliptic Operators and Compact Groups. Lect. Notes Math. 401. Zbl. 297. 58009

    Google Scholar 

  • Atiyah, M.F. (1976): Elliptic operators, discrete groups and von Neumann algebras. Astérisque 32–33, 43–72. Zbl. 323. 58015

    Google Scholar 

  • Atiyah, M.F., Bott, R.A. (1967): A Lefschetz fixed point formula for elliptic complexes. I. Ann. Math., II. Ser. 87, 347–407. Zbl. 161, 432

    Google Scholar 

  • Atiyah, M.F., Bott, R.A. (1984): The moment map and equivariant cohomology. Topology 23, 1–28. Zbl. 521. 58025

    Google Scholar 

  • Atiyah, M.F., Bott, R.A., Patody, V.K. (1973): On the heat equation and the index theorem. Invent. Math. 19, 279–330. Zbl. 257. 58008.

    Google Scholar 

  • Atiyah, M.F., Bott, R.A., Patody, V.K. Errata (1975) 29, 277–280. Zbl. 301. 58018

    Google Scholar 

  • Atiyah, M.F., Schmid, W. (1977): A geometric construction of the discrete series for semi-simple Lie groups. Invent. Math. 42, 1–62. Zbl. 373. 22001

    Google Scholar 

  • Atiyah, M.F., Segal, G.B. (1968): The index of elliptic operators. II. Ann. Math., II. Ser. 87, 531–545. Zbl. 164, 242

    Google Scholar 

  • Atiyah, M.F., Singer, I.M. (1968a): The index of elliptic operators. I. Ann. Math., II. Ser. 87, 484–530. Zbl. 164, 240

    Google Scholar 

  • Atiyah, M.F., Singer, I.M. (1968b): The index of elliptic operators. III. Ann. Math., II. Ser. 87, 546–604. Zbl. 164, 243

    Google Scholar 

  • Atiyah, M.F., Singer, I.M. (1971a): The index of elliptic operators. IV. Ann. Math., H. Ser. 93, 119–138. Zbl. 212, 286

    Google Scholar 

  • Atiyah, M.F., Singer, I.M. (1971b): The index of elliptic operators. V. Ann. Math., II. Ser. 93, 139–149. Zbl. 212, 286

    Google Scholar 

  • Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D. (1978): Defor-mation theory and quantization. Ann. Phys. 111, 61–110. Zbl. 377. 53024

    Google Scholar 

  • Berline, N., Vergne, M. (1985): The equivariant index and Kirillov’s character for-mula. Am. J. Math. 109, 1159–1190. Zbl. 604. 58046

    Google Scholar 

  • Bismut, J.-M. (1984): The Atiyah-Singer theorems: a probabilistic approach. I: The index theorem. II: The Lefschetz fixed point formulas. J. Funct. Anal. 57, 56–99; 329–348. I: Zbl. 538.58033. I I: Zbl. 556. 58027

    Google Scholar 

  • Booss, B. (1977): Topologie und Analysis Eine Einführung in die Atiyah-Singer Indexformel. Springer, Berlin Heidelberg New York. Zbl. 364. 58017

    Google Scholar 

  • Bott, R. (1967a): Lectures on K(X). Harvard Univ. preprint (W.A. Benjamin, 1969.Zbl. 194, 239 )

    Google Scholar 

  • Bott, R. (196713): A residue formula for holomorphic vector fields. J. Differ. Geom. 1, 311–330. Zbl. 179, 288

    Google Scholar 

  • Boutet de Monvel, L. (1971): Boundary problems for pseudodifferential operators. Acta Math. 126, 11–51. Zbl. 206, 394

    Google Scholar 

  • Boutet de Monvel, L. (1979): On the index of Toeplitz operators of several complex variables. Invent. Math. 50, 249–272. Zbl. 398. 47018

    Google Scholar 

  • Boutet de Monvel, L., Guillemin, V. (1989): The spectral theory of Toeplitz operators. Ann. Math. Stud. 99. Zbl. 469. 47021

    Google Scholar 

  • Brenner, A.V., Shubin, M.A. (1981): The Atiyah-Bott-Lefschetz theorem for manifolds with boundary. Funkts. Anal. Prilozh. 15, no. 4, 67–68.

    MathSciNet  MATH  Google Scholar 

  • Brenner, A.V., Shubin, M.AEnglish transl.: Funct. Anal. Appl. 15, 286–287 (1982). Zbl. 483. 58015

    Google Scholar 

  • Breuer, M. ( 1968, 1969): Fredholm theories in von Neumann algebras. I, II. Math. Ann. 178, 243–254; 180, 313–325. I: Zb1.162,187, II: Zbl. 175, 441

    Google Scholar 

  • Chern, S.S. (1956): Complex Manifolds. University Press, Chicago. Zbl. 74, 303

    Google Scholar 

  • Coburn, L.A., Moyer, R.D., Singer, I.M. (1973): The C’-algebras of almost periodic pseudo-differential operators. Acta Math. 130, 279–307. Zbl. 263. 47042

    Google Scholar 

  • Colin de Verdier, Y. (1979): Sur le spectre des opérateurs à bicaractéristiques périodiques. Comment. Math. Hely. 54, 508–522. Zbl. 459. 58014

    Google Scholar 

  • Connes, A., Moscovici, H. (1982): The L2-index theorem for homogeneous spaces of Lie groups. Ann. Math., II Ser. 115, 291–330. Zbl. 515. 58031

    Google Scholar 

  • Dupont, J.L. (1978): Curvature and Characteristic Classes. Lect. Notes Math. 640. Zbl. 373. 57009

    Google Scholar 

  • Emch, G. (1972): Algebraic methods in Statistical Mechanics and Quantum Field Theory. Wiley-Interscience, New York. Zbl. 235. 46085

    Google Scholar 

  • Fedosov, B.V. (1974): Analytic index formulae. Tr. Mosk. Mat. 0.-va 30, 159–240.

    MathSciNet  MATH  Google Scholar 

  • Fedosov, B.V. English transl.: Trans. Mosc. Math. Soc. 30, 159–240 (1976). Zbl. 349. 58006

    Google Scholar 

  • Fedosov, B.V. (1978): A periodicity theorem in the symbol algebra. Mat. Sb., Nov. Ser. 105, 431–462.

    MathSciNet  Google Scholar 

  • Fedosov, B.V. English transi.: Math. USSR, Sb. 34, 382–410 (1978). Zbl. 391. 47033

    Google Scholar 

  • Fedosov, B.V. (1986): Quantisation and the index. Dokl. Akad. Nauk SSSR 291, 82–86.

    MathSciNet  Google Scholar 

  • Fedosov, B.V. English transl.: Sov. Phys., Dokl. 31, 877–878 (1986). Zbl. 635. 58019

    Google Scholar 

  • Fedosov, B.V. (1989): The index theorem in the algebra of quantum observables. Dokl. Akad. Nauk SSSR 305, 835–838.

    MathSciNet  Google Scholar 

  • Fedosov, B.V. English transl.: Sov. Phys., Dokl. 34, 319321 (1989)

    Google Scholar 

  • Fedosov, B.V., Shubin, M.A. (1978): The index of stochastic elliptic operators. I, II. Mat. Sb., Nov. Ser. 106, 108–140, 455–483.

    Google Scholar 

  • Fedosov, B.V., Shubin, M.A. English transl.: Math. USSR, Sb. 34, 671–699 (1978); 35, 131–156 (1979). I: Zbl. 409.47030, II: Zbl. 392. 58010

    Google Scholar 

  • Forster, O. (1977): Riemannsche Flächen. Springer-Verlag, Berlin Heidelberg New York. Zbl. 381. 30021

    Google Scholar 

  • Gel’fand, I.M. (1960): On elliptic equations. Usp. Mat. Nauk 15, No. 3, 121–132.

    Google Scholar 

  • Gel’fand, I.M. English transi.: Russ. Math. Surv. 15, No. 3, 113–123 (1960). Zbl. 95, 78

    Google Scholar 

  • Getzler, E. (1983): Pseudodifferential operators on supermanifold and the AtiyahSinger index theorem. Commun. Math. Phys. 92, 163–178. Zbl. 543. 58026

    Google Scholar 

  • Gohberg, I.C., Krein, M.G. (1965): Introduction to the Theory of Linear Nonselfadjoint Operators in a Hilbert Space. Nauka, Moscow.

    Google Scholar 

  • Gohberg, I.C., Krein, M.G. English transi.: Am. Math. Soc., Providence 1969. Zbl. 138, 78

    Google Scholar 

  • Hörmander, L. (1971): Fourier integral operators. I. Acta Math. 127, 79–183. Zbl. 212, 466

    Google Scholar 

  • Hörmander, L. (1979): The Weyl calculus of pseudodifferential operators. Commun. Pure Appl. Math. 32, 358–443. Zbl. 388. 47032

    Google Scholar 

  • Hörmander, L. (1984–1985): The Analysis of Linear Partial Differential Operators. 1–4. Springer-Verlag, Berlin Heidelberg New York. I,II: Zbl. 521.35001/2, III: Zbl. 601.35001, IV: Zbl. 612. 35001

    Google Scholar 

  • Husemöller, D. (1966): Fibre Bundles. McGraw-Hill, New York. Zbl. 144, 448

    Google Scholar 

  • Karasev, M.V., Maslov, V.P. (1984): Asymptotic and geometric quantisation. Usp.Mat. Nauk 39, No. 6, 115–173.

    MathSciNet  Google Scholar 

  • Karasev, M.V., Maslov, V.P. English transi.: Russ. Math. Surv. 39, No. 6,133–205 (1984). Zbl. 588. 58031

    Google Scholar 

  • Kirillov, A.A. (1972): Elements of Representation Theory. Nauka, Moscow.

    Google Scholar 

  • Kirillov, A.A. English transi.: Springer-Verlag, Berlin Heidelberg New York 1976. Zbl. 264. 22011

    Google Scholar 

  • Kirillov, A.A. (1985): Geometric Quantisation. (Itogi Nauki Tekh., Sovr. Probl. Mat., Fundam. Napr..4.) Moscow, 141–178.

    Google Scholar 

  • Kirillov, A.A. English transl. in: Encycl. Math. Sci..4, Springer-Verlag, Berlin Heidelberg New York 1988, 137–172. Zbl. 591. 58014

    Google Scholar 

  • Lecomte, P., De Wilde, M. (1983): Existence of star products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7, 487–496. Zbl. 526. 58023

    Google Scholar 

  • Lichnerowicz, A. (1955): Théorie Globale des Connexions et des Groups d’Holonomie. Edizioni Cremonese, Rome. Zbl. 116, 391

    Google Scholar 

  • Palais, R.S. (1965): Seminar on the Atiyah-Singer Index Theorem. Princeton Univ. Press, Princeton. Zbl. 137, 170

    Google Scholar 

  • Rempel, S., Schulze, B.-W. (1982): Index Theory of Elliptic Boundary Problems. Akademie-Verlag, Berlin. Zbl. 504. 35002

    Google Scholar 

  • Schwarz, A.S. (Shvarts) (1981): Elliptic Operators in quantum field theory. (Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 17.) Moscow, 113–173.

    Google Scholar 

  • Schwarz, A.S. English transl.: J. Sov. Math. 21, 551–601 (1983). Zbl. 482. 35080

    Google Scholar 

  • Shubin, M.A. (1978): Pseudodifferential Operators and Spectral Theory. Nauka, Moscow. Zbl. 451. 47064

    Google Scholar 

  • Shubin, M.A. (1979): Spectral theory and the index of elliptic operators with almost periodic coefficients. Usp. Mat. Nauk 34, No. 2, 95–135.

    MathSciNet  MATH  Google Scholar 

  • Shubin, M.A. English transl.: Russ. Math. Surv. 34, No. 2, 109–157 (1979). Zbl. 431. 47027

    Google Scholar 

  • Springer, G. (1957): Introduction to Riemann Surfaces. Addison-Wesley, Reading, Mass. Zbl. 78, 66

    Google Scholar 

  • Treves, F. (1980): Introduction to Pseudodifferential and Fourier Integral Operators. Plenum, New York. Zbl. 453. 47027

    Google Scholar 

Download references

Authors
  1. B. V. Fedosov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Northeastern University, 02115, Boston, MA, USA

    M. A. Shubin

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Fedosov, B.V. (1996). Index Theorems. In: Shubin, M.A. (eds) Partial Differential Equations VIII. Encyclopaedia of Mathematical Sciences, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48944-0_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-48944-0_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-48946-4

  • Online ISBN: 978-3-642-48944-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation