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Book Information:
Authors:
Dusa McDuff and
Dietmar Salamon
Title:
-holomorphic curves and symplectic topology
Additional book information:
Colloquium Publications, vol. 52,
Amer. Math. Soc., Providence, RI,
2004,
xii+669 pp.,
ISBN 0-8218-3485-1,
US$99.00$
Vladimir Arnol′d, Sur une propriété topologique des applications globalement canoniques de la mécanique classique, C. R. Acad. Sci. Paris 261 (1965), 3719–3722 (French). MR 193645
Yuri Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002), no. 3, 441–483. MR 1946550, DOI 10.1007/s002220200212
C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnol′d, Invent. Math. 73 (1983), no. 1, 33–49. MR 707347, DOI 10.1007/BF01393824
Boris Dubrovin, Geometry of $2$D topological field theories, Integrable systems and quantum groups (Montecatini Terme, 1993) Lecture Notes in Math., vol. 1620, Springer, Berlin, 1996, pp. 120–348. MR 1397274, DOI 10.1007/BFb0094793
Yakov Eliashberg, Invariants in contact topology, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 327–338. MR 1648083
Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume (2000), 560–673. GAFA 2000 (Tel Aviv, 1999). MR 1826267, DOI 10.1007/978-3-0346-0425-3_{4}
Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228
Andreas Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813. MR 948771, DOI 10.1002/cpa.3160410603
9. Fukaya, K., Oh, Y-G., Ono, K., Ohta, H., Lagrangian intersection Floer theory - anomaly and obstruction, to be published in Int. Press.
Alexander Givental, The mirror formula for quintic threefolds, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, vol. 196, Amer. Math. Soc., Providence, RI, 1999, pp. 49–62. MR 1736213, DOI 10.1090/trans2/196/04
11. Givental, A., Semisimple Frobenius structures at higher genus, arXiv math. AG/0008067.
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), no. 1-2, 25–38. MR 1059642, DOI 10.1017/S0308210500024549
Maxim Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335–368. MR 1363062, DOI 10.1007/978-1-4612-4264-2_{1}2
François Lalonde and Jean-Claude Sikorav, Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comment. Math. Helv. 66 (1991), no. 1, 18–33 (French). MR 1090163, DOI 10.1007/BF02566634
Dusa McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc. 3 (1990), no. 3, 679–712. MR 1049697, DOI 10.1090/S0894-0347-1990-1049697-8
Dusa McDuff and Dietmar Salamon, $J$-holomorphic curves and quantum cohomology, University Lecture Series, vol. 6, American Mathematical Society, Providence, RI, 1994. MR 1286255, DOI 10.1090/ulect/006
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
Leonid Polterovich, The geometry of the group of symplectic diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. MR 1826128, DOI 10.1007/978-3-0348-8299-6
Clifford Henry Taubes, Seiberg Witten and Gromov invariants for symplectic $4$-manifolds, First International Press Lecture Series, vol. 2, International Press, Somerville, MA, 2000. Edited by Richard Wentworth. MR 1798809
Paul H. Rabinowitz, Periodic solutions of a Hamiltonian system on a prescribed energy surface, J. Differential Equations 33 (1979), no. 3, 336–352. MR 543703, DOI 10.1016/0022-0396(79)90069-X
- 1.
- Arnold, V. I., Sur une proprieté topologique des applications globalement canonique de la mécanique classique, C.R. Acad. Sci. Paris 261 (1965), 3719-3722. MR 0193645 (33:1861)
- 2.
- Chekanov, Yu., Differential Algebra of Legendrian Links, Invent. Math. 150 (2002), 441-483. MR 1946550 (2003m:53153)
- 3.
- Conley, C. C., Zehnder, E., The Birkhoff-Lewis fixed-point theorem and a conjecture of V. I. Arnold, Invent. Math., 73 (1983), 3-49. MR 0707347 (85e:58044)
- 4.
- Dubrovin, B., Geometry of 2D topological field theories, Springer Lecture Notes in Math. 1620 (1996), 120-348. MR 1397274 (97d:58038)
- 5.
- Eliashberg, Y., Invariants in Contact Topology, Proc. ICM-98, Berlin, Doc. Math., 327-338. MR 1648083 (2000a:57068)
- 6.
- Eliashberg, Y., Givental, A., Hofer, H., Introduction to Symplectic Field Theory in Geom. Funct. Anal., Special volume, Part II (2000), 560-673. MR 1826267 (2002e:53136)
- 7.
- Floer, A., Morse theory for Lagrangian intersections, J. Diff. Geom., 28 (1988), 513-547. MR 0965228 (90f:58058)
- 8.
- Floer, A., The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math 41 (1988), 775-813. MR 0948771 (89g:58065)
- 9.
- Fukaya, K., Oh, Y-G., Ono, K., Ohta, H., Lagrangian intersection Floer theory - anomaly and obstruction, to be published in Int. Press.
- 10.
- Givental, A., The mirror formula for quintic threefolds, Amer. Math. Soc. Transl. Ser. 2, 196 (1999), 49-62. MR 1736213 (2000j:14083)
- 11.
- Givental, A., Semisimple Frobenius structures at higher genus, arXiv math. AG/0008067.
- 12.
- Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. MR 0809718 (87j:53053)
- 13.
- Hofer, H., On the topological properties of symplectic maps, Proc. Royal Soc. Edinburgh 115 (1990), 25-38. MR 1059642 (91h:58042)
- 14.
- Kontsevich, M., Enumeration of rational curves via torus action, in The moduli space of curves, Progress in Math. 129, Birkhäuser, 1995. MR 1363062 (97d:14077)
- 15.
- Lalonde, F., Sikorav, J.-C., Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comm. Math. Helv. 66 (1991), 18-33. MR 1090163 (92f:58060)
- 16.
- McDuff, D., The structure of rational and ruled symplectic
-manifolds, J. Amer. Math. Soc. 3 (1990), 679-712. MR 1049697 (91k:58042)
- 17.
- McDuff, D., Salamon, D.,
-holomorphic curves and quantum cohomology, Amer. Math. Soc. Univ. Lect. Ser. 6, 1994. MR 1286255 (95g:58026)
- 18.
- Ozsváth, P., Szabó, Z., Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), 1027-1158. MR 2113019 (2006b:57016)
- 19.
- Polterovich, L., The Geometry of the Group of Symplectic Diffeomorphisms, Birkhäuser, 2001. MR 1826128 (2002g:53157)
- 20.
- Taubes, C. H., The Seiberg-Witten and Gromov invariants for symplectic
-manifolds, First Internat. Press Lect. Ser. 2, International Press, Somerville, MA, 2000. MR 1798809 (2002j:53115)
- 21.
- Weinstein, A., On the hypothesis of Rabinowitz' periodic orbit theorems, J. Diff. Eq. 33 (1979), 353-358. MR 0543704 (81a:58030b)
Review Information:
Reviewer:
Yakov Eliashberg
Affiliation:
Stanford University
Email:
eliash@math.stanford.edu
Journal:
Bull. Amer. Math. Soc.
44 (2007), 309-315
Published electronically:
December 1, 2006
Review copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
