Geometry of determinants of elliptic operators

  • Maxim Kontsevich
  • Simeon Vishik
Part of the Progress in Mathematics book series (PM, volume 131)

Abstract

D.B. Ray and I.M. Singer invented zeta-regularized determinants for positive definite elliptic pseudo-differential operators (PDOs) of positive orders acting in the space of smooth sections of a finite-dimensional vector bundle E over a closed finite-dimensional manifold M ([RS1], [RS2]).

Keywords

Manifold Simeon 

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References

  1. [APS]
    Atiyah, M.F., Patodi, V.K., Singer, I.M., Spectral asymmetry and Riemannian geometry, I. Math. Proc. Cambridge Phil. Soc. (1975) 77, 43–69MathSciNetCrossRefMATHGoogle Scholar
  2. [Fr]
    Friedlander, L., Ph.D Thesis, Dept. Math. MIT, 1989Google Scholar
  3. [Hö]
    Hörmander, L., The analysis of linear partial differential operators, I. Grundl. math. Wiss. 256, Springer-Verlag: Berlin, Heidelberg, 1983Google Scholar
  4. [Kas]
    Kassel, C., Le residue non commutatif (d’apres M. Wodzicki). Semin. Bourbaki, 41eme ann. 1988–89, Exp. 708 (1989)Google Scholar
  5. [KV]
    Kontsevich, M., Vishik, S., Determinants of elliptic pseudo- differential operators, preprint Max-Planck-Institut fur Mathematik 1994, MPI/94-30, 156 pp. (Submitted to GAFA)Google Scholar
  6. [KrKh]
    Kravchenko, O.S., Khesin, B.A., Central extension of the Lie algebra of (pseudo-) differential symbols, Funct. Anal, and its Appl 25: 2 (1991), 83–85MathSciNetGoogle Scholar
  7. [Li]
    Lidskii, V.B., Nonselfadjoint operators with a trace, Dokl. Akad. Nauk SSSR 125 (1959), 485–487MathSciNetGoogle Scholar
  8. [R]
    Radul, A.O., Lie algebras of differential operators, their central extensions, and W-algebras, Funct. Anal. 25: 1 (1991), 33–49MathSciNetGoogle Scholar
  9. [RSI]
    Ray, D.B., Singer, I.M., R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145–210MathSciNetCrossRefMATHGoogle Scholar
  10. [RS2]
    Ray, D.B., Singer, I.M., Analytic torsion for complex manifolds. Ann. Math. 98(1973), 154–177Google Scholar
  11. [Re]
    Retherford, J.R., Hilbert space: Compact operators and the trace theorem, London Math. Soc. Student Texts 27, Cambridge Univ. Press, 1993Google Scholar
  12. [Sch]
    Schwarz, A.S., The partition function of a degenerate functional, Commun. Math. Phys. 67 (1979), 1–16CrossRefMATHGoogle Scholar
  13. [Wol]
    Wodzicki, M., Local invariants of spectral asymmetry, Invent. Math. 75 (1984), 143–178MathSciNetCrossRefMATHGoogle Scholar
  14. [Wo2]
    Wodzicki, M., “Noncommutative residue. Chap. I. Fundamentals”, In: K-theory, arithmetic and geometry, Lect. Notes Math. 1289 (1987), Springer-Verlag: Heidelberg, New York, 320–399Google Scholar

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Maxim Kontsevich
    • 1
    • 2
  • Simeon Vishik
    • 1
    • 3
  1. 1.Max-Planck-Institut für MathematikBonnGermany
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of Mathematics 038-16Temple UniversityPhiladelphiaUSA

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