A distributional approach to asymptotics of the Spectral Action

dc.contributorPaycha, Sylvie
dc.contributorNeira Jiménez, Carolina
dc.creatorLópez Castaño, Juan Daniel
dc.date.accessioned2021-02-24T15:39:02Z
dc.date.available2021-02-24T15:39:02Z
dc.date.created2021-02-24T15:39:02Z
dc.date.issued2020-08-18
dc.identifierLópez-Castaño, J. D. (2020) A distributional approach to asymptotics of the Spectral Action. Master thesis, Universidad Nacional de Colombia.
dc.identifierhttps://repositorio.unal.edu.co/handle/unal/79294
dc.description.abstractLa acción espectral es el concepto natural y apropiado para hablar de una acción en el espacio de triplas espectrales, y fue introducido por primera vez por Chamseddine y Connes en 1997. Después de incluir definiciones y resultados que conciernen a la teoría de Cesàro para distribuciones y análisis asintótico, discutimos la expansión asintótica de la acción espectral en el sentido distribucional para una tripla espectral conmutativa, siguiendo a Estrada, Gracia-Bondía y Várilly.
dc.description.abstractThe spectral action is the natural and appropriate notion of an action on the space of spectral triples, and it was introduced by Chamseddine and Connes in 1997. After including some definitions and results concerning the Cesàro theory of distributions and asymptotic analysis, we discuss the asymptotic expansion of the spectral action in the distributional sense for a commutative spectral triple following Estrada, Gracia-Bondía and Várilly.
dc.languageeng
dc.publisherBogotá - Ciencias - Maestría en Ciencias - Matemáticas
dc.publisherDepartamento de Matemáticas
dc.publisherUniversidad Nacional de Colombia - Sede Bogotá
dc.relationN. Berline, E. Getzler, and M. Vergne. Heat Kernels and Dirac Operators. Grundlehren Text Editions. Springer Berlin Heidelberg, 2003. ISBN: 9783540200628. URL: https://books.google.com.co/books?id=\_e2FjvLbO94C (see p. 81).
dc.relationAli H. Chamseddine and Alain Connes. “The spectral action principle”. In: Comm. Math. Phys. 186.3 (1997), pp. 731–750. ISSN: 0010-3616. DOI: 10.1007/s002200050126. URL: https://doi.org/10.1007/s002200050126 (see pp. 9, 67, 71).
dc.relationAli H. Chamseddine, Alain Connes, and Matilde Marcolli. “Gravity and the standard model with neutrino mixing”. In: Adv. Theor. Math. Phys. 11.6 (2007), pp. 991–1089. ISSN: 1095-0761. URL: http://projecteuclid.org/euclid.atmp/1198095373 (see p. 9).
dc.relationAlain Connes and Matilde Marcolli. Noncommutative geometry, quantum fields and motives. Vol. 55. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008, pp. xxii+785. ISBN: 978-0-8218-4210-2 (see pp. 9, 64, 71).
dc.relationAlain Connes and Matilde Marcolli. Noncommutative geometry, quantum fields and motives. Vol. 55. American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008, pp. xxii+785. ISBN: 978-0-8218-4210-2 (see pp. 10, 69, 75).
dc.relationAlain Connes. “On the spectral characterization of manifolds”. In: J. Noncommut. Geom. 7.1 (2013), pp. 1–82. ISSN: 1661-6952. DOI:10.4171/JNCG/108. URL: https://doi.org/10.4171/JNCG/108 (see p. 66).
dc.relationAlain Connes. Noncommutative geometry. Academic Press, Inc., San Diego, CA, 1994, pp. xiv+661. ISBN: 0-12-185860-X (see pp. 9, 63).
dc.relationR. Estrada, J. M. Gracia-Bondía, and J. C. Várilly. “On summability of distributions and spectral geometry”. In: Comm. Math. Phys. 191.1 (1998), pp. 219–248. ISSN: 0010-3616. DOI: 10.1007/s002200050266. URL: https://doi.org/10.1007/s002200050266 (see pp. 9, 10, 19, 30, 49, 54, 58, 75).
dc.relationM. Eckstein and B. Iochum. Spectral Action in Noncommutative Geometry. SpringerBriefs in Mathematical Physics. Springer International Publishing, 2018. ISBN: 9783319947884. URL: https://books.google.com.co/books?id=nnWADwAAQBAJ (see p. 15).
dc.relationRicardo Estrada and Ram P. Kanwal. A distributional approach to asymptotics. Second. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Theory and applications. Birkhäuser Boston, Inc., Boston, MA, 2002, pp. xvi+451. ISBN: 0-8176-4142-4. DOI: 10.1007/978-0-8176-8130-2. URL: https://doi.org/10.1007/978-0-8176-8130-2 (see pp. 9, 13, 19, 34, 37, 39).
dc.relationR Estrada, R. P Kanwal, and Michael James Lighthill. “A distributional theory for asymptotic expansions”. In: Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 428.1875 (1990), pp. 399–430. DOI: 10.1098/rspa.1990.0041. eprint: https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1990.0041. URL: https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1990.0041 (see pp. 34, 48, 58).
dc.relationRicardo Estrada. “The Cesàro behaviour of distributions”. In: R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454.1977 (1998), pp. 2425–2443. ISSN: 1364-5021. DOI: 10.1098/rspa.1998.0265. URL: https://doi.org/10.1098/rspa.1998.0265 (see pp. 9, 13, 26, 28, 34).
dc.relationBoris V. Fedosov et al. “The noncommutative residue for manifolds with boundary”. In: J. Funct. Anal. 142.1 (1996), pp. 1–31. ISSN: 0022-1236. DOI: 10.1006/jfan.1996.0142. URL: https://doi.org/10.1006/jfan.1996.0142 (see pp. 77, 79, 80).
dc.relationG.B. Folland. Real Analysis: Modern Techniques and Their Applications. A Wiley-Interscience publication. Wiley, 1999. ISBN: 9780471317166. URL: https://books.google.com.co/books?id=uPkYAQAAIAAJ (see p. 31).
dc.relationWentao Fan, Farzad Fathizadeh, and Matilde Marcolli. “Spectral action for Bianchi type-IX cosmological models”. In: J. High Energy Phys. 10 (2015), 085, front matter+28. ISSN: 1126-6708. DOI:10.1007/JHEP10(2015)085. URL: https://doi.org/10.1007/JHEP10(2015)085 (see p. 9).
dc.relationG.B. Folland. Advanced Calculus. Featured Titles for Advanced Calculus Series. Prentice Hall, 2002. ISBN: 9780130652652. URL: https://books.google.com.co/books?id=iatzQgAACAAJ (see p. 26).
dc.relationJosé M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa. Elements of noncommutative geometry. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Boston, Inc., Boston, MA, 2001, pp. xviii+685. ISBN: 0-8176-4124-6. DOI: 10.1007/978-1-4612-0005-5. URL: https://doi.org/10.1007/978-1-4612-0005-5 (see pp. 9, 66, 81, 84, 85).
dc.relationPeter B. Gilkey. Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Second. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995, pp. x+516. ISBN: 0-8493-7874-4 (see pp. 9, 68, 69).
dc.relationA. Grossmann, G. Loupias, and E. M. Stein. “An algebra of pseudodifferential operators and quantum mechanics in phase space”. In: Ann. Inst. Fourier (Grenoble) 18.fasc., fasc. 2 (1968), 343–368, viii (1969). ISSN: 0373-0956. URL: http://www.numdam.org/item?id=AIF_1968__18_2_343_0 (see pp. 34, 35).
dc.relationGerd Grubb. Distributions and operators. Vol. 252. Graduate Texts in Mathematics. Springer, New York, 2009, pp. xii+461. ISBN: 978-0-387-84894-5 (see pp. 9, 13, 19, 20, 21, 25, 35, 77, 78, 79).
dc.relationG. H. Hardy. Divergent series. With a preface by J. E. Littlewood and a note by L. S. Bosanquet, Reprint of the revised (1963) edition. Éditions Jacques Gabay, Sceaux, 1992, pp. xvi+396. ISBN: 2-87647-131-0 (see pp. 13, 15, 16, 18, 19, 87).
dc.relationB. Iochum, C. Levy, and D. V. Vassilevich. “Global and local aspects of spectral actions”. In: J. Phys. A 45.37 (2012), pp. 374020, 19. ISSN: 1751-8113. DOI: 10.1088/1751-8113/45/37/374020. URL: https://doi.org/10.1088/1751-8113/45/37/374020 (see p. 9).
dc.relationAnthony W. Knapp. Representation Theory of Semisimple Groups: An Overview Based on Examples (PMS-36). REV - Revised. Princeton University Press, 1986. ISBN: 9780691084015. URL: http://www.jstor.org/stable/j.ctt1bpm9sn (see p. 44).
dc.relationH. Blaine Lawson Jr. and Marie-Louise Michelsohn. Spin geometry. Vol. 38. Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1989, pp. xii+427. ISBN: 0-691-08542-0 (see pp. 81, 84)
dc.relationMatilde Marcolli, Elena Pierpaoli, and Kevin Teh. “The spectral action and cosmic topology”. In: Comm. Math. Phys. 304.1 (2011), pp. 125–174. ISSN: 0010-3616. DOI: 10.1007/s00220-011-1211-3. URL: https://doi.org/10.1007/s00220-011-1211-3 (see p. 9).
dc.relationKarl-Hermann Neeb. “On differentiable vectors for representations of infinite dimensional Lie groups”. In: Journal of Functional Analysis 259.11 (2010), pp. 2814 –2855. ISSN: 0022-1236. DOI: https://doi.org/10.1016/j.jfa.2010.07.020. URL: http://www.sciencedirect.com/science/article/pii/S0022123610003095 (see p. 46).
dc.relationS. Paycha. Regularised Integrals, Sums, and Traces: An Analytic Point of View. University lecture series. American Mathematical Society, 2012. ISBN: 9780821890356. URL: https://books.google.com.co/books?id=ut0tnQAACAAJ (see p. 56).
dc.relationMichael Reed and Barry Simon. Methods of modern mathematical physics. I. Second. Functional analysis. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980, pp. xv+400. ISBN: 0-12-585050-6 (see pp. 41, 42, 43, 44).
dc.relationMairi Sakellariadou. “Noncommutative geometry spectral action as a framework for unification: introduction and phenomenological/cosmological consequences”. In: Internat. J. Modern Phys. D 20.5 (2011), pp. 785–804. ISSN: 0218-2718. DOI: 10.1142/S021827181101913X. URL: https://doi.org/10.1142/S021827181101913X (see p. 9).
dc.relationHeikkilä Seppo and Talvila Erik. “Distributions, their primitives and integrals with applications to distributional differential equations”. In: Dynamic Systems and Applications 22 (June 2013), pp. 207–249 (see p. 26).
dc.relationR. T. Seeley. “Complex powers of an elliptic operator”. In: Proc. Symp. Pure Math. 10 (1967), pp. 288–307 (see p. 53).
dc.relationK. Yosida. Functional Analysis. Classics in Mathematics. Springer Berlin Heidelberg, 2012. ISBN: 9783642618598. URL: https://books.google.com.co/books?id=yj4mBQAAQBAJ (see p. 46).
dc.relationA. H. Zemanian. “The Distributional Laplace and Mellin Transformations”. In: SIAM Journal on Applied Mathematics 14.1 (1966), pp. 41–59. DOI: 10.1137/0114004. eprint: https://doi.org/10.1137/0114004. URL: https://doi.org/10.1137/0114004 (see p. 72).
dc.rightsAtribución-NoComercial 4.0 Internacional
dc.rightsAcceso abierto
dc.rightshttp://creativecommons.org/licenses/by-nc/4.0/
dc.rightsinfo:eu-repo/semantics/openAccess
dc.rightsDerechos reservados - Universidad Nacional de Colombia
dc.titleA distributional approach to asymptotics of the Spectral Action
dc.typeOtro


Este ítem pertenece a la siguiente institución