The notion of dimension in geometry and algebra

Manin, Yuri

doi:10.1090/S0273-0979-06-01081-0

Author: Yuri I. Manin
Journal: Bull. Amer. Math. Soc. 43 (2006), 139-161
MSC (2000): Primary 14H10, 14N10
DOI: https://doi.org/10.1090/S0273-0979-06-01081-0
Published electronically: February 8, 2006
MathSciNet review: 2216108
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Abstract: 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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1007/s000290050042
Alain Connes, Cyclic cohomology, quantum group symmetries and the local index formula for ${\rm SU}_q(2)$, J. Inst. Math. Jussieu 3 (2004), no. 1, 17–68. MR 2036597, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/0019-3577%2894%2990015-9
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 https://doi.org/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 https://doi.org/10.1016/0393-0440%2894%2900032-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) Springer, Berlin, 1973, pp. 69–190. Lecture Notes in Mathematics, Vol. 350. MR 0447119
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 https://doi.org/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 https://doi.org/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
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 https://doi.org/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, W. H. Freeman and Co., San Francisco, Calif., 1982. Schriftenreihe für den Referenten. [Series for the Referee]. 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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, 1972) Springer, Berlin, 1973, pp. 191–268. Lecture Notes in Math., Vol. 350 (French). MR 0404145
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 https://doi.org/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 0175892
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; Librairie Gauthier-Villars, Paris, 1957, pp. 261–289 (French). MR 0108765
Maxim Vybornov, Constructible sheaves on simplicial complexes and Koszul duality, Math. Res. Lett. 5 (1998), no. 5, 675–683. MR 1666864, DOI https://doi.org/10.4310/MRL.1998.v5.n5.a10
André Weil, Elliptic functions according to Eisenstein and Kronecker, Springer-Verlag, Berlin-New York, 1976. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 88. MR 0562289