Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The notion of dimension in geometry and algebra
HTML articles powered by AMS MathViewer

by Yuri I. Manin
Bull. Amer. Math. Soc. 43 (2006), 139-161
Published electronically: February 8, 2006


This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared.
  • Greg W. Anderson, Cyclotomy and an extension of the Taniyama group, Compositio Math. 57 (1986), no. 2, 153–217. MR 827351
  • Michael Atiyah, Commentary on the article of Yu. I. Manin: “New dimensions in geometry” [Workshop Bonn 1984 (Bonn, 1984), 59–101, Springer, Berlin, 1985; MR0797416 (87j:14030)], Workshop Bonn 1984 (Bonn, 1984) Lecture Notes in Math., vol. 1111, Springer, Berlin, 1985, pp. 103–109. MR 797417, DOI 10.1007/BFb0084586
  • A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
  • [Br]Br T. Bridgeland. Stability conditions on triangulated categories. Preprint math.AG/0212237
  • Alain Connes, $C^{\ast }$ algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 13, A599–A604 (French, with English summary). MR 572645
  • Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
  • Alain Connes, Geometry from the spectral point of view, Lett. Math. Phys. 34 (1995), no. 3, 203–238. MR 1345552, DOI 10.1007/BF01872777
  • Alain Connes, Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996), no. 1, 155–176. MR 1441908
  • Alain Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function, Selecta Math. (N.S.) 5 (1999), no. 1, 29–106. MR 1694895, DOI 10.1007/s000290050042
  • Alain Connes, Cyclic cohomology, quantum group symmetries and the local index formula for $\textrm {SU}_q(2)$, J. Inst. Math. Jussieu 3 (2004), no. 1, 17–68. MR 2036597, DOI 10.1017/S1474748004000027
  • Alain Connes and Dirk Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), no. 1, 249–273. MR 1748177, DOI 10.1007/s002200050779
  • [CoMar1]CoMar1 A. Connes, M. Marcolli. Quantum statistical mechanics of $\mathbf {Q}$–lattices. (From Physics to Number Theory via Noncommutative Geometry, Part I). Preprint math.NT/0404128
  • Alain Connes and Matilde Marcolli, Renormalization and motivic Galois theory, Int. Math. Res. Not. 76 (2004), 4073–4091. MR 2109986, DOI 10.1155/S1073792804143122
  • A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), no. 2, 174–243. MR 1334867, DOI 10.1007/BF01895667
  • Caterina Consani and Matilde Marcolli, Noncommutative geometry, dynamics, and $\infty$-adic Arakelov geometry, Selecta Math. (N.S.) 10 (2004), no. 2, 167–251. MR 2080121, DOI 10.1007/s00029-004-0369-3
  • [DaLSSV]DaLSSV L. Da̧browski, G. Landi, A. Sitarz, W. van Suijlekom, J.C. Varilly. The Dirac operator on $SU_{q}(2)$. Preprint math.QA/0411609.
  • Christopher Deninger, Local $L$-factors of motives and regularized determinants, Invent. Math. 107 (1992), no. 1, 135–150. MR 1135468, DOI 10.1007/BF01231885
  • Christopher Deninger, Motivic $L$-functions and regularized determinants, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 707–743. MR 1265547, DOI 10.1090/pspum/055.1/1265547
  • Christopher Deninger, Some analogies between number theory and dynamical systems on foliated spaces, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 163–186. MR 1648030
  • Christopher Deninger, A note on arithmetic topology and dynamical systems, Algebraic number theory and algebraic geometry, Contemp. Math., vol. 300, Amer. Math. Soc., Providence, RI, 2002, pp. 99–114. MR 1936368, DOI 10.1090/conm/300/05144
  • Michael R. Douglas, Dirichlet branes, homological mirror symmetry, and stability, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 395–408. MR 1957548
  • M. J. Shai Haran, The mysteries of the real prime, London Mathematical Society Monographs. New Series, vol. 25, The Clarendon Press, Oxford University Press, New York, 2001. MR 1872029
  • [He]He T. L. Heath. The thirteen books of Euclid’s Elements. Translation, Introduction and Commentary. Cambridge UP, 1908.
  • W. Kalau and M. Walze, Gravity, non-commutative geometry and the Wodzicki residue, J. Geom. Phys. 16 (1995), no. 4, 327–344. MR 1336738, DOI 10.1016/0393-0440(94)00032-Y
  • [KapSm]KapSm M. Kapranov, A. Smirnov. Cohomology determinants and reciprocity laws: number field case. Unpublished.
  • Nicholas M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin-New York, 1973, pp. 69–190. MR 447119
  • Dirk Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998), no. 2, 303–334. MR 1633004, DOI 10.4310/ATMP.1998.v2.n2.a4
  • Michel L. Lapidus and Carl Pomerance, Fonction zêta de Riemann et conjecture de Weyl-Berry pour les tambours fractals, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 6, 343–348 (French, with English summary). MR 1046509
  • Michel L. Lapidus and Machiel van Frankenhuysen, Complex dimensions of fractal strings and oscillatory phenomena in fractal geometry and arithmetic, Spectral problems in geometry and arithmetic (Iowa City, IA, 1997) Contemp. Math., vol. 237, Amer. Math. Soc., Providence, RI, 1999, pp. 87–105. MR 1710790, DOI 10.1090/conm/237/1710790
  • Michel L. Lapidus and Machiel van Frankenhuysen, Fractal geometry and number theory, Birkhäuser Boston, Inc., Boston, MA, 2000. Complex dimensions of fractal strings and zeros of zeta functions. MR 1726744, DOI 10.1007/978-1-4612-5314-3
  • J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math. (2) 153 (2001), no. 1, 191–258. MR 1826413, DOI 10.2307/2661374
  • J. Lewis and D. Zagier, Period functions and the Selberg zeta function for the modular group, The mathematical beauty of physics (Saclay, 1996) Adv. Ser. Math. Phys., vol. 24, World Sci. Publ., River Edge, NJ, 1997, pp. 83–97. MR 1490850
  • Benoit B. Mandelbrot, The fractal geometry of nature, Schriftenreihe für den Referenten. [Series for the Referee], W. H. Freeman and Co., San Francisco, CA, 1982. MR 665254
  • [Ma]Ma D. Yu. Manin. Personal communication.
  • Yu. I. Manin, New dimensions in geometry, Uspekhi Mat. Nauk 39 (1984), no. 6(240), 47–73 (Russian). MR 771098
  • Yu. I. Manin, Three-dimensional hyperbolic geometry as $\infty$-adic Arakelov geometry, Invent. Math. 104 (1991), no. 2, 223–243. MR 1098608, DOI 10.1007/BF01245074
  • Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228 (1995), 4, 121–163. Columbia University Number Theory Seminar (New York, 1992). MR 1330931
  • [Ma4]Ma4 Yu. Manin. Von Zahlen und Figuren. Preprint math.AG/0201005.
  • Yu. I. Manin, Real multiplication and noncommutative geometry (ein Alterstraum), The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 685–727. MR 2077591
  • Yuri I. Manin and Matilde Marcolli, Continued fractions, modular symbols, and noncommutative geometry, Selecta Math. (N.S.) 8 (2002), no. 3, 475–521. MR 1931172, DOI 10.1007/s00029-002-8113-3
  • Dieter H. Mayer, Continued fractions and related transformations, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989) Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, pp. 175–222. MR 1130177
  • Barry Mazur, Notes on étale cohomology of number fields, Ann. Sci. École Norm. Sup. (4) 6 (1973), 521–552 (1974). MR 344254
  • Masanori Morishita, On certain analogies between knots and primes, J. Reine Angew. Math. 550 (2002), 141–167. MR 1925911, DOI 10.1515/crll.2002.070
  • A. Polishchuk and A. Schwarz, Categories of holomorphic vector bundles on noncommutative two-tori, Comm. Math. Phys. 236 (2003), no. 1, 135–159. MR 1977884, DOI 10.1007/s00220-003-0813-9
  • [Po1]Po1 A. Polishchuk. Noncommutative two–tori with real multiplication as noncommutative projective varieties. Preprint math.AG/0212306. [Po2]Po2 A. Polishchuk. Classification of holomorphic vector bundles on noncommutative two–tori. Preprint math.QA/0308136.
  • Marc A. Rieffel, von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981), no. 4, 403–418. MR 639575, DOI 10.1007/BF01465863
  • Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin-New York, 1973, pp. 191–268 (French). MR 404145
  • A. L. Smirnov, Hurwitz inequalities for number fields, Algebra i Analiz 4 (1992), no. 2, 186–209 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 2, 357–375. MR 1182400
  • [Sm2]Sm2 A. Smirnov. Letters to Yu. Manin of Sept. 29 and Nov. 29, 2003.
  • Christophe Soulé, Les variétés sur le corps à un élément, Mosc. Math. J. 4 (2004), no. 1, 217–244, 312 (French, with English and Russian summaries). MR 2074990, DOI 10.17323/1609-4514-2004-4-1-217-244
  • John Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 288–295. MR 175892
  • J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques, Établissements Ceuterick, Louvain, 1957, pp. 261–289 (French). MR 108765
  • Maxim Vybornov, Constructible sheaves on simplicial complexes and Koszul duality, Math. Res. Lett. 5 (1998), no. 5, 675–683. MR 1666864, DOI 10.4310/MRL.1998.v5.n5.a10
  • André Weil, Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 88, Springer-Verlag, Berlin-New York, 1976. MR 562289
Similar Articles
  • Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 14H10, 14N10
  • Retrieve articles in all journals with MSC (2000): 14H10, 14N10
Bibliographic Information
  • Yuri I. Manin
  • Affiliation: Northwestern University, Evanston, Illinois, USA
  • Address at time of publication: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
  • MR Author ID: 190766
  • Received by editor(s): April 24, 2005
  • Published electronically: February 8, 2006
  • Additional Notes: Based on the talks delivered at the AMS sectional meeting, Northwestern University, October 2004; and Blyth Lectures, University of Toronto, November 2004
  • © Copyright 2006 Yuri I. Manin
  • Journal: Bull. Amer. Math. Soc. 43 (2006), 139-161
  • MSC (2000): Primary 14H10, 14N10
  • DOI:
  • MathSciNet review: 2216108