Bulletin of the American Mathematical Society

Perturbations, deformations, and variations (and ``near-misses") in geometry, physics, and number theory

Author(s): B. Mazur
Journal: Bull. Amer. Math. Soc. 41 (2004), 307-336.
MSC (2000): Primary 11Jxx, 14Bxx, 32Gxx
Posted: April 16, 2004
Retrieve article in: PDF

[A1]: Arnold, V.I.: Catastrophe Theory (transl. Wassermann, G.S.), Springer-Verlag, 3rd revised and expanded edition (1992) MR 93h:58018
[A2]: Arnold, V.I., Topological Invariants of Plane Curves and Caustics, University Lecture Series, 5, American Mathematical Society (1994) MR 95h:57003
[B-F-F-L-S]: Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation Theory and Quantization. Ann. Phys. 111 (1978), 61-151 MR 58:14737a
[B-C-D-T]: Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over $\bf Q$ : wild -adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843-939 MR 2002d:11058
[Ca-F]: Cattaneo, A., Felder, G.: A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), no. 3, 591-611 MR 2002b:53141
[Co]: Connes, A.: A Short Survey of Noncommutative Geometry. J. Math. Phys. 41 (2000), 3832-3866 MR 2001m:58016
[Cr-M]: Crauder, B., Miranda, R.: Quantum cohomology of rational surfaces, pp. 33-79 in The Moduli Space of Curves, Eds: Dijkgraff, R., Faber, C., van der Geer, G., Birkhäuser (1995) MR 97i:14033
[De]: Deligne, P: Déformations de l'algèbre d'une variété symplectique: comparaison entre Fedosov et De Wilde, Lecomte. (French), Selecta Math. 1 (1995), no. 4, 667-697 MR 97a:58068
[D-L]: Dewilde, M., Lecomte, P.: Existence of Star-Products and of Formal Deformations of the Poisson Lie Algebra of Arbitrary Symplectic Manifolds. Lett. Math. Phys. 7 (1983), 487-496 MR 85j:17021
[Di]: Dijkgraaf, R.: Mirror symmetry and elliptic curves, pp. 149-164 in The Moduli Space of Curves, Eds: Dijkgraff, R., Faber, C., van der Geer, G., Birkhäuser (1995) MR 96m:14072
[Do]: Doran, C.: Deformation Theory: An Historical Annotated Bibliography, www.math.columbia.edu/ doran/Hist%20Ann%20Bib.pdf
[E]: Elkies, N.: Rational points near curves and small nonzero $\vert x^3-y^2\vert$ via lattice reduction, Lecture Notes in Computer Science 1838 (proceedings of ANTS-4, 2000; W. Bosma, ed.), Springer, Berlin, 2000, 33-63 (math.NT/0005139 on the arXiv) MR 2002g:11035
[E-L-S-V]: Ekedahl, T., Lando, S., Shapiro, M., Vainshtein, A.: On Hurwitz numbers and Hodge integrals, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 12, 1175-1180 MR 2001b:14083
[F]: Fedosov, B. V.: A Simple Geometrical Construction of Deformation Quantization. J. Diff. Geom. 40 (1994), 213-238 MR 95h:58062
[G]: Gibson, C. G., Singular points of smooth mappings, Research Notes in Mathematics, Pitman (1979) MR 80j:58011
[G-L-S]: Greuel, G.-M., Lossen C., and Shustin, E.: Plane curves of minimal degree with prescribed singularities, Invent. Math. 133 (1998) 539-580 MR 99g:14035
[H]: Harris, J.: On the Severi problem, Invent. Math. 84 (1986) 445-461 MR 87f:14012
[K-Z]: Kaneko, M., Zagier, D.: A generalized Jacobi Theta Function and Quasi-modular Forms, pp. 165-172, in The Moduli Space of Curves, Eds: Dijkgraff, R., Faber, C., van der Geer, G., Birkhäuser (1995) MR 96m:11030
[Ka]: Kani, E.: The Number of Genus Covers of an Elliptic Curve, preprint; see: www.ams.org/amsmtgs/210_abstracts/991-14-302.pdf
[K-M]: Kapovich, M., Millson, J. J.: On the deformation theory of representations of fundamental groups of closed hyperbolic 3-manifolds, Topology, Vol. 35, No. 4 (1996) 1085-1106 MR 97h:57029
[Ko 1]: Kontsevich, M.: Enumeration of rational curves via torus actions, pp. 335-368, in The Moduli Space of Curves, Eds: Dijkgraff, R., Faber, C., van der Geer, G., Birkhäuser (1995) MR 97d:14077
[Ko 2]: Kontsevich, M.: Deformation Quantization of Poisson Manifolds, I. Preprint q-alg/9709040 (September 1997)
[La]: Lagrange, J.-L.: Mécanique Analytique, Gauthier-Villars, Paris (1888)
[Lo]: Loeser, F.: Déformations de courbes planes (d'aprés Severi et Harris), Séminaire Bourbaki, 1986/87, Astérisque, No. 152-153 (1987) 4 187-205 MR 89d:14038
[M]: Mazur, B.: An ``infinite fern" in the deformation space of Galois representations, Collect. Math. 48, 1-2 (1997) 155-193 MR 98g:11053
[M-F]: Modular Forms and Fermat's Last Theorem, Eds: Cornell, G., Silverman, J., Stevens, G., Springer (1997) MR 99k:11004
[M-R-Y]: McLellan, B., Roth, M., Yui, N.: Generating Functions Counting Simply Ramified Covers, preprint, Nov. 2003
[P]: The Mathematical Heritage of Henri Poincaré, Volume 39, Part 2, Proceedings of Symposia in Pure Mathematics, American Mathematical Society (1983) MR 85h:00010b
[R]: Riemann, B.: Theorie der Abel'schen Functionen, pp. 88-144 in The Collected Works of Bernhard Riemann, Ed: Weber, H., Dover (1953)
[Ru]: Rudd, R.: The string partition function for QCD on the Torus, hep-th/9407176
[S]: Severi, F.: Vorlesung über Algebraische Geometrie, Teubner (1921)
[Q-F]: Quantum Fields and Strings: A course for mathematicians, vols. I, II, Eds: Deligne, P. et al., A.M.S and I.A.S. (1999) MR 2000e:81010
[T]: Thom, R.: Structural stability and morphogenesis, W. A. Benjamin, Inc., Reading, MA (1976) MR 58:7722b
[Z]: Zariski, O.: Collected Papers, volume III: Topology of curves and surfaces, and special topics in the theory of algebraic varieties, Eds: Artin, M., and Mazur, B., M.I.T. Press (1978) MR 58:21346c

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 11Jxx, 14Bxx, 32Gxx

B. Mazur
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138-2901
Email: mazur@math.harvard.edu

DOI: 10.1090/S0273-0979-04-01024-9
PII: S 0273-0979(04)01024-9
Received by editor(s): February 19, 2004
Posted: April 16, 2004
Dedicated: In memory of René Thom
Additional Notes: I wish to thank Curt McMullen and Noam Elkies for helpful comments on an early draft, and Stephanie Yang and Ivan Petrakiev for providing the illustrations
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS