| Integrable Systems in the Realm of Algebraic Geometry 2001 Edition Contributor(s): Vanhaecke, Pol (Author) |
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ISBN: 3540423370 ISBN-13: 9783540423379 Publisher: Springer OUR PRICE: $52.24 Product Type: Paperback Published: July 2001 Annotation: This book treats the general theory of Poisson structures and integrable systems on affine varieties in a systematic way. Special attention is drawn to algebraic completely integrable systems. Several integrable systems are constructed and studied in detail and a few applications of integrable systems to algebraic geometry are worked out. In the second edition some of the concepts in Poisson geometry are clarified by introducting Poisson cohomology; the Mumford systems are constructed from the algebra of pseudo-differential operators, which clarifies their origin; a new explanation of the multi Hamiltonian structure of the Mumford systems is given by using the loop algebra of sl(2); and finally Goedesic flow on SO(4) is added to illustrate the linearizatin algorith and to give another application of integrable systems to algebraic geometry. |
| Additional Information |
| BISAC Categories: - Mathematics | Geometry - Algebraic - Medical - Mathematics | Differential Equations - General |
| Dewey: 516.353 |
| LCCN: 2001049076 |
| Series: Lecture Notes in Computer Science |
| Physical Information: 0.58" H x 6.14" W x 9.21" (0.86 lbs) 264 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 2. Divisors and line bundles ........................ 99. 2.1. Divisors .............................. 99. 2.2. Line bundles ............................ 100. 2.3. Sections of line bundles ....................... 101. 2.4. The Riemann-Roch Theorem ..................... 103. 2.5. Line bundles and embeddings in projective space ............ 105. 2.6. Hyperelliptic curves ......................... 106. 3. Abelian varieties ............................ 108. 3.1. Complex tori and Abelian varieties .................. 108. 3.2. Line bundles on Abelian varieties ................... 109. 3.3. Abelian surfaces .......................... 111. 4. Jacobi varieties ............................. 114. 4.1. The algebraic Jacobian ....................... 114. 4.2. The analytic/transcendental Jacobian ................. 114. 4.3. Abel's Theorem and Jacobi inversion ................. 119. 4.4. Jacobi and Kummer surfaces ..................... 121. 5. Abelian surfaces of type (1,4) ....................... 123. 5.1. The generic case .......................... 123. 5.2. The non-generic case ........................ 124. V. Algebraic completely integrable Hamiltonian systems ........ 127. 1. Introduction .............................. 127. 2. A.c.i. systems ............................. 129. 3. Painlev analysis for a.c.i, systems .................... 135. 4. The linearization of two-dkmensional a.e.i, systems ............. 138. 5. Lax equations ............................. 140. VI. The Mumford systems ..................... 143. 1. Introduction .............................. 143. 2. Genesis ................................ 145. 2.1. The algebra of pseudo-differential operators .............. 145. 2.2. The matrix associated to two commuting operators ........... 146. 2.3. The inverse construction ....................... 150. 2.4. The KP vector fields ........................ 152. ix 3. Multi-Hamiltonian structure and symmetries ................ 155. 3.1. The loop algebra 9(q ........................ 155. 3.2. Reducing the R-brackets and the vector field ............. 157. 4. The odd and the even Mumford systems .................. 161. 4.1. The (odd) Mumford system ..................... 161. 4.2. The even Mumford system ...................... 163. |