The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. Its mathematical applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov-Witten invariants and quantum cohomology--two important ingredients of the mirror symmetry conjecture. <P>This book establishes the fundamental theorems of the subject in full and rigorous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associativity of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: There are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of $J$-holomorphic curves in four dimensional manifolds.
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"The book offers a systematic treatment of one of the important new fields in mathematics and should be available in every library." ---- EMS Newsletter
"Written by two top specialists on this topic, it will be useful to all those interested in this fascinating domain. Last but not least the graphical conditions offered by the American Mathematical Society are excellent." ---- Zentralblatt Math
"Written by two top specialists on this topic, it will be useful to all those interested in this fascinating domain. Last but not least the graphical conditions offered by the American Mathematical Society are excellent." ---- Zentralblatt Math
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1 of 5 people found the following review helpful:
2.0 out of 5 stars
but indispensible,
By Dash Riprock "math student" (New York, NY USA) - See all my reviews
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This review is from: $J$-holomorphic Curves and Symplectic Topology (Colloquium Publications (Amer Mathematical Soc)) (Hardcover)
an invaluable reference for users. but for a first stab at j-holomorphic curves and applications in symplectic topology one might want to try the little version instead (same authors, title contains the words "quantum cohomology") - which as a bonus is available for free on at least one of the authors' website (i think).
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First Sentence:
The theory of J-holomorphic curves, introduced by Gromov in 1985, has profoundly influenced the study of symplectic geometry, and now permeates almost all its aspects. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
fixed marked points, flic proof, simple stable maps, stable neaps, quantum cup product, spherical homology class, stable leaps, gluing theorem, complex structure oil, nioduli space, adjunction inequality, evaluation neap, vortex equations, smooth cutoff function, standard complex structure, quantum cohomology, finite dimensional reduction, pairwise distinct points, local intersection number, monotone case, gluing map, connecting orbits, spectral invariants, divisor equation, symplectic action Key Phrases - Capitalized Phrases (CAPs): (learn more)
Gromov Witten, Fubini Study, Gronlov Witten, Cauchy Riemann, Grontov Witten, Micallef White, Grolnov Witten, Gronrov Witten, Grothendieck Knudsen, Grotuov Witten, Calabi Yau, Groinov Witten, Grornov Witten, Groutov Witten, Ionel Parker New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The theory of J-holomorphic curves, introduced by Gromov in 1985, has profoundly influenced the study of symplectic geometry, and now permeates almost all its aspects. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
fixed marked points, flic proof, simple stable maps, stable neaps, quantum cup product, spherical homology class, stable leaps, gluing theorem, complex structure oil, nioduli space, adjunction inequality, evaluation neap, vortex equations, smooth cutoff function, standard complex structure, quantum cohomology, finite dimensional reduction, pairwise distinct points, local intersection number, monotone case, gluing map, connecting orbits, spectral invariants, divisor equation, symplectic action Key Phrases - Capitalized Phrases (CAPs): (learn more)
Gromov Witten, Fubini Study, Gronlov Witten, Cauchy Riemann, Grontov Witten, Micallef White, Grolnov Witten, Gronrov Witten, Grothendieck Knudsen, Grotuov Witten, Calabi Yau, Groinov Witten, Grornov Witten, Groutov Witten, Ionel Parker New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 9
books:
See all 9 books this book cites
- Torus Actions on Symplectic Manifolds (Progress in Mathematics) by Michèle Audin in Back Matter
- Introduction To Commutative Algebra by Michael Atiyah in Back Matter
- Symplectic Geometry (London Mathematical Society Lecture Note Series) by Dietmar Salamon in Back Matter
- Differential Equations (2nd Edition) by John Polking in Back Matter
- Morse Homology (Progress in Mathematics (Birkhauser Boston)) by Matthias Schwarz in Back Matter
See all 9 books this book cites
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