Applications of noncommutative geometry in function theory and mathematical physics

Sergeev, Armen

doi:10.1090/mosc/307

Applications of noncommutative geometry in function theory and mathematical physics
HTML articles powered by AMS MathViewer

by Armen Sergeev

Trans. Moscow Math. Soc. 2020, 123-167

DOI: https://doi.org/10.1090/mosc/307

Published electronically: March 15, 2021

PDF | Request permission

Abstract:

We review some applications of noncommutative geometry to function theory and mathematical physics. In the first case we discuss relations between the spaces of real variables and operator algebras. In the second case we deal with quantization of universal Techmüller space and quantum Hall effect.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
J. T. Ashkroft and N. D. Mermin, Solid state physics, Philadelphia: Saunders Co., 1976.
F. A. Berezin and M. A. Shubin, The Schrödinger equation, Mathematics and its Applications (Soviet Series), vol. 66, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1983 Russian edition by Yu. Rajabov, D. A. Leĭtes and N. A. Sakharova and revised by Shubin; With contributions by G. L. Litvinov and Leĭtes. MR 1186643, DOI 10.1007/978-94-011-3154-4
J. Bellissard, A. van Elst, and H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35 (1994), no. 10, 5373–5451. Topology and physics. MR 1295473, DOI 10.1063/1.530758
F. A. Berezin, The method of second quantization, Pure and Applied Physics, Vol. 24, Academic Press, New York-London, 1966. Translated from the Russian by Nobumichi Mugibayashi and Alan Jeffrey. MR 0208930
Rufus Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11–25. MR 556580
A. L. Carey, K. C. Hannabuss, V. Mathai, and P. McCann, Quantum Hall effect on the hyperbolic plane, Comm. Math. Phys. 190 (1998), no. 3, 629–673. MR 1600480, DOI 10.1007/s002200050255
Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
H. M. Farkas and I. Kra, Riemann surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992. MR 1139765, DOI 10.1007/978-1-4612-2034-3
José 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. MR 1789831, DOI 10.1007/978-1-4612-0005-5
Svante Janson and Thomas H. Wolff, Schatten classes and commutators of singular integral operators, Ark. Mat. 20 (1982), no. 2, 301–310. MR 686178, DOI 10.1007/BF02390515
Y. Kordyukov, V. Mathai, and M. Shubin, Equivalence of spectral projections in semiclassical limit and a vanishing theorem for higher traces in $K$-theory, J. Reine Angew. Math. 581 (2005), 193–236. MR 2132676, DOI 10.1515/crll.2005.2005.581.193
L. D. Landau and E. M. Lifshitz, Course of theoretical physics. Vol. 5: Statistical physics, Second revised and enlarged edition, Pergamon Press, Oxford-Edinburgh-New York, 1968. Translated from the Russian by J. B. Sykes and M. J. Kearsley. MR 0242449
B. Laughlin, Quantized Hall conductivity in two dimensions, Phys. Rev. 1984. Vol. B23. P. 5232.
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
Matilde Marcolli and Varghese Mathai, Towards the fractional quantum Hall effect: a noncommutative geometry perspective, Noncommutative geometry and number theory, Aspects Math., E37, Friedr. Vieweg, Wiesbaden, 2006, pp. 235–261. MR 2327308, DOI 10.1007/978-3-8348-0352-8_{1}2
Subhashis Nag and Dennis Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle, Osaka J. Math. 32 (1995), no. 1, 1–34. MR 1323099
M. A. Naĭmark, Normed rings, P. Noordhoff N. V., Groningen, 1959. Translated from the first Russian edition by Leo F. Boron. MR 0110956
Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210, DOI 10.1007/978-0-387-21681-2
S. C. Power, Hankel operators on Hilbert space, Research Notes in Mathematics, vol. 64, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 666699
H. M. Reimann, Ordinary differential equations and quasiconformal mappings, Invent. Math. 33 (1976), no. 3, 247–270. MR 409804, DOI 10.1007/BF01404205
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Graeme Segal, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys. 80 (1981), no. 3, 301–342. MR 626704
Armen Sergeev, Lectures on universal Teichmüller space, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2014. MR 3243752, DOI 10.4171/141
A. G. Sergeev, Geometric quantization of loop spaces, Moscow: Steklov Institute, 2009 (in Russian).
Armen Sergeev, Kähler geometry of loop spaces, MSJ Memoirs, vol. 23, Mathematical Society of Japan, Tokyo, 2010. MR 2654515, DOI 10.1142/e023
A. G. Sergeev, Quantum calculus and ideals in the algebra of compact operators, Tr. Mat. Inst. Steklova 306 (2019), no. Matematicheskaya Fisika i Prilozheniya, 227–234 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 306 (2019), no. 1, 212–219. MR 4040777, DOI 10.4213/tm4004
A. Sergeev, Quantum differentials and function spaces, Accepted for publication in the Proc. of the Conference devoted to N. Vasilevski. Birkhäuser, 2020.
A. Sergeev, Quantum Hall effect and non-commutative geoemtry, Accepted for publication in the Proc. of the Conference dedicated to the memory of B. Sternin. Birkhäuser, 2020.
A. G. Sergeev, In search of infinite-dimensional Kähler geometry, Uspekhi Mat. Nauk 75 (2020), no. 2(452), 133–184 (Russian, with Russian summary); English transl., Russian Math. Surveys 75 (2020), no. 2, 321–367. MR 4081970, DOI 10.4213/rm9919
A. Sergeev, Universal Teichmüller space as a non-trivial example of infinite-dimensional complex manifolds, Handbook of Teichmüller Theory. Vol. 7. Berlin: European Math. Society, 2020. P. 195–213. (Lect. Math. Theor. Phys.; Vol. 30.)
A. G. Sergeev, Magnetic Bloch theory and noncommutative geometry, Tr. Mat. Inst. Steklova 279 (2012), no. Analiticheskie i Geometricheskie Voprosy Kompleksnogo Analiza, 193–205 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 279 (2012), no. 1, 181–193. MR 3086764, DOI 10.1134/s0081543812080123
A. G. Sergeev, Fractional Hall effect from the point of view of noncommutative geometry, Proc. Seminar on vector and tensor analysis. Vol. 28. P. 250–270. Moscow: MSU, 2012 (in Russian).
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
D. J. Thouless, Quantized Hall conductance, particle transport and topological invariants, XIIIth international colloquium on group theoretical methods in physics (College Park, Md., 1984) World Sci. Publishing, Singapore, 1984, pp. 319–323. MR 815710
Jingbo Xia, Geometric invariants of the quantum Hall effect, Comm. Math. Phys. 119 (1988), no. 1, 29–50. MR 968479