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Large-$ N $ Limits of Spaces and Structures

  • *Corresponding author: Ambar N. Sengupta

    *Corresponding author: Ambar N. Sengupta
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  • For a sequence of spaces $ X_N $, with topological, algebraic or measure-theoretic structures, we show how a large-$ N $ limit $ X_\infty $ with corresponding structures is obtained. For example, when each space is a topological group $ G_N $, such as $ G_N = U(N) $, a limiting group $ G_\infty $ with topology results. Using the Weil-Kodaira construction, for compact topological groups $ G_N $ equipped with normalized Haar measures, we obtain a topological structure on $ G_\infty $ that also makes the group operations continuous. When each $ G_N $ is a Lie group we describe a Lie algebra associated to $ G_\infty $.

    Mathematics Subject Classification: Primary: 28E04, 54J05; Secondary: 28C10, 28C20.

    Citation:

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  • [1] I. Alam, An introduction to the probabilistic method through the Lovász Local Lemma, arXiv: 1909.11078.
    [2] I. Alam, Limiting probability measures, J. Log. Anal. 12 (2020), 35 pp. doi: 10.4115/jla.2020.12.1.
    [3] S. AlbeverioR. H$\phi$egh-KrohnJ. Fenstad and  T. Lindstr$\phi$mNonstandard methods in stochastic analysis and mathematical physics, Academic Press, Inc., Orlando, FL, 1986. 
    [4] P. Bankston, A Survey of Ultraproduct Constructions in General Topology, arXiv: math/9709203v1.
    [5] A. Borovik and M. G. Katz, Who gave you the Cauchy-Weierstrass tale? The dual history of rigorous calculus, Found. Sci.. 17 (2012), 245-276. doi: 10.1007/s10699-011-9235-x.
    [6] H. Cartan, Théorie des filtres, C. R. Acad. Sci. Paris, 205 (1937), 595-598. 
    [7] H. Cartan, Théorie des ultrafiltres., C. R. Acad. Sci. Paris, 205 (1937), 777-778. 
    [8] F. Destrempes and A. Sengupta, Configurations of points in sets of positive measure and in Baire sets of second category, Fund. Math., 133 (1989), 155-159.  doi: 10.4064/fm-133-2-155-159.
    [9] B. K. DriverF. GabrielB. C. Hall and T. Kemp, The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces, Commun. Math. Phys., 352 (2017), 967-978.  doi: 10.1007/s00220-017-2857-2.
    [10] T. FrayneD. Scott and A. Tarski, Reduced Products., Notices, Amer. Math. Soc., 5 (1958), 673-674. 
    [11] T. E. Frayne, A. Morel and D. Scott, Reduced direct products, Fund. Math.. 51 (1962), 195-228. doi: 10.4064/fm-51-3-195-228.
    [12] I. Goldbring, Nonstandard hulls of locally exponential Lie algebras, J. Log. Anal., 1 (2009), 25 pp. doi: 10.4115/jla.2009.1.5.
    [13] I. Goldbring, Nonstandard hulls of locally uniform groups, Fund. Math., 220 (2013), 93-118.  doi: 10.4064/fm220-2-1.
    [14] I. Goldbring, Ultrafilters Throughout Mathematics, American Mathematical Society, Providence, RI (2022). doi: 10.1090/gsm/220.
    [15] D. Gross and A. Matytsin, Some properties of large-N two-dimensional Yang-Mills theory, Nuclear Phys. B. 437 (1995), 541-584. doi: 10.1016/0550-3213(94)00570-5.
    [16] L. GrossC. King and A. Sengupta, Two-dimensional Yang-Mills theory via stochastic differential equations, Ann. Phys., 194 (1989), 65-112.  doi: 10.1016/0003-4916(89)90032-8.
    [17] P. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950
    [18] H. J. Keisler, A survey of ultraproducts, Logic, Methodology And Philos. Sci. (Proc. 1964 Internat. Congr.). pp. 112-126 (1965)
    [19] K. Kodaira, über die Beziehung zwischen den Massen und den Topologien in einer Gruppe, Proc. Phys.-Math. Soc. Japan (3). 23 (1941), 67-119.
    [20] T. Lévy, The master field on the plane, Astérisque., ix+201 (2017)
    [21] P. A. Loeb, A combinatorial analog of Lyapunov's theorem for infinitesimally generated atomic vector measures, Proc. Amer. Math. Soc.. 39 (1973), 585-586. doi: 10.2307/2039598.
    [22] P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc.. 211 (1975), 113-122. doi: 10.2307/1997222.
    [23] J. Łoś, Quelques Remarques, Thèorémes et Problémes sur les Classes Dèfinissables d'Algébres, in Mathematical Interpretation Of Formal Systems, 1955.
    [24] R. Parikh, A nonstandard theory of topological groups, in Applications Of Model Theory To Algebra, Analysis, And Probability (Internat. Sympos., Pasadena, Calif., 1967), 1969.
    [25] A. Robinson, Non-Standard Analysis, North-Holland Publishing Co., Amsterdam (1966)
    [26] P. Samuel, Ultrafilters and compactification of Uniform Spaces. (ProQuest LLC, Ann Arbor, MI, 1947), Ph.D thesis, Princeton University, 1947.
    [27] A. Sengupta, Quantum gauge theory on compact surfaces, Ann. Physics. 221 (1993), 17-52. doi: 10.1006/aphy.1993.1002.
    [28] I. M. Singer, On the master field in two dimensions, in Functional Analysis On The Eve Of The 21st Century, Vol. 1 (New Brunswick, NJ, 1993). 131 (1995), 263-281.
    [29] T. Tao, Ultraproducts as a Bridge Between Discrete and Continuous Analysis, https://terrytao.wordpress.com/2013/12/07/ultraproducts-as-a-bridge-between-discrete-and-continuous-analysis/.
    [30] A. Weil, Sur les groupes topologiques et les groupes mesures., C. R. Acad. Sci., Paris. 202 (1936), 1147-1149.
    [31] A. Weil, Sur les espaces à structure uniforme et sur la topologie générale., Oeuvres Scientifiques/Collected Papers. I., (2014), 1926-1951.
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