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Motivic information

  • Matilde Marcolli
Article

Abstract

We introduce notions of information/entropy and information loss associated to exponentiable motivic measures. We show that they satisfy appropriate analogs to the Khinchin-type properties that characterize information loss in the context of measures on finite sets.

Notes

Acknowledgements

The author is partially supported by NSF Grant DMS-1707882 and NSERC Grants RGPIN-2018-04937 and RGPAS-2018-522593 and by the Perimeter Institute for Theoretical Physics.

References

  1. 1.
    Almkvist, G.: Endomorphisms of finitely generated projective modules over a commutative ring. Ark. Mat. 11, 263–301 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baez, J.C., Fritz, T., Leinster, T.: A characterization of entropy in terms of information loss. Entropy 13(11), 1945–1957 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bittner, F.: The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140(4), 1011–1032 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bloch, S.: Algebraic K-theory and crystalline cohomology. Inst. Hautes Études Sci. Publ. Math. 47(1977), 187–268 (1978)zbMATHGoogle Scholar
  5. 5.
    Borisov, L.: The class of the affine line is a zero divisor in the Grothendieck ring (2014). arXiv:1412.6194 (preprint)
  6. 6.
    Bost, J.B., Connes, A.: Hecke algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory. Sel. Math. 1(3), 411–457 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives, Colloquium Publications, vol. 55. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  8. 8.
    Connes, A., Marcolli, M.: Quantum statistical mechanics of \({{\mathbb{Q}}}\) -lattices. In: Frontiers in Number Theory, Physics, and Geometry, I,. Springer, New York, pp. 269–347 (2006)Google Scholar
  9. 9.
    Faddeev, D.K.: On the concept of entropy of a finite probabilistic scheme. Uspehi Mat. Nauk 11(1), 227–231 (1956)Google Scholar
  10. 10.
    Gillet, H., Soulé, C.: Descent, motives and K-theory. J. Reine Angew. Math. 478, 127–176 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gusein-Zade, S.M., Luego, I., Melle-Hernández, A.: A power structure over the Grothendieck ring of varieties. Math. Res. Lett. 11, 49–57 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hesselholt, L.: The big de Rham-Witt complex. Acta Math. 214, 135–207 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Julia, B.: Statistical theory of numbers. In: Luck, J.M., Moussa, P., Waldschmidt, M. (eds.) Number Theory and Physics, pp. 276–293. Springer, New York (1990)CrossRefGoogle Scholar
  14. 14.
    Kapranov, M.: The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups. arXiv:math/0001005
  15. 15.
    Khinchin, A.I.: Mathematical Foundations of Information Theory. Dover, New York (1957)zbMATHGoogle Scholar
  16. 16.
    Larsen, M., Lunts, V.: Rationality criteria for motivic zeta functions. Compos. Math. 140(6), 1537–1560 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Leinster, T.: The categorical origins of entropy, lecture at “Topological and Geometric Structures of Information”. CIRM, Luminy (2017). http://forum.cs-dc.org/topic/575/tom-leinster-the-categorical-origins-of-entropy
  18. 18.
    Loeb, D.E., Rota, G.C.: Formal power series of logarithmic type. Adv. Math. 75(1), 1–118 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Looijenga, E.: Motivic measures, Séminaire Bourbaki, vol. 1999/2000. Astérisque 276, 267–297 (2002)Google Scholar
  20. 20.
    Marcolli, M., Thorngren, R.: Thermodynamic semirings. J. Noncommutative Geom. 8(2), 337–392 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marcolli, M.: Information algebras and their applications. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information, Lecture Notes in Computer Science, vol. 9389. Springer, New York, pp. 271–276 (2015)Google Scholar
  22. 22.
    Martin, N.: The class of the affine line is a zero divisor in the Grothendieck ring: an improvement. C. R. Math. Acad. Sci. Paris 354(9), 936–939 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mustaţǎ, M.: Zeta Functions in Algebraic geometry, Lecture Notes. http://www-personal.umich.edu/~mmustata (preprint)
  24. 24.
    Poonen, B.: The Grothendieck ring of varieties is not a domain. Math. Res. Lett. 9(4), 493–497 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ramachandran, N.: Zeta functions, Grothendieck groups, and the Witt ring. Bull. Sci. Math. Soc. Math. Fr. 139(6), 599–627 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ramachandran, N., Tabuada, G.: Exponentiable motivic measures. J. Ramanujan Math. Soc. 30(4), 349–360 (2015)MathSciNetGoogle Scholar
  27. 27.
    Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication, vol. 1949. University of Illinois Press, Champaign (1998)zbMATHGoogle Scholar
  28. 28.
    Shen, Y.: The \({\rm GL}_{\rm n}\)-Connes-Marcolli Systems. arXiv:1609.08727
  29. 29.
    Spector, D.: Supersymmetry and the Möbius inversion function. Commun. Math. Phys. 127, 239–252 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tsfasman, M., Vladut, S.G.: Algebraic-Geometric Codes. Kluwer, Dordrecht (1991)CrossRefzbMATHGoogle Scholar

Copyright information

© Unione Matematica Italiana 2018

Authors and Affiliations

  1. 1.University of TorontoTorontoCanada
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  3. 3.California Institute of TechnologyPasadenaUSA

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