LEADER 04070nam 22005171 450 001 9910811556803321 005 20240516003438.0 010 $a0-19-154584-8 035 $a(CKB)2550000001130185 035 $a(MiAaPQ)EBC3055739 035 $a(Au-PeEL)EBL3055739 035 $a(CaPaEBR)ebr10782467 035 $a(CaONFJC)MIL530280 035 $a(OCoLC)861200296 035 $a(EXLCZ)992550000001130185 100 $a20110805d2011 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $2rdacontent 182 $2rdamedia 183 $2rdacarrier 200 10$aRiemann surfaces /$fSimon Donaldson 205 $a1st ed. 210 1$aOxford ;$aNew York :$cOxford University Press,$d2011. 215 $a1 online resource (301 pages) $cillustrations 225 0$aOxford graduate texts in mathematics ;$v22 311 $a0-19-960674-9 311 $a1-299-99029-0 320 $aIncludes bibliographical references (pages 282-283) and index. 327 $aCover -- Contents -- PART I: PRELIMINARIES -- 1 Holomorphic functions -- 1.1 Simple examples: algebraic functions -- 1.2 Analytic continuation: differential equations -- Exercises -- 2 Surface topology -- 2.1 Classification of surfaces -- 2.2 Discussion: the mapping class group -- Exercises -- PART II: BASIC THEORY -- 3 Basic definitions -- 3.1 Riemann surfaces and holomorphic maps -- 3.2 Examples -- Exercises -- 4 Maps between Riemann surfaces -- 4.1 General properties -- 4.2 Monodromy and the Riemann Existence Theorem -- Exercises -- 5 Calculus on surfaces -- 5.1 Smooth surfaces -- 5.2 de Rham cohomology -- 5.3 Calculus on Riemann surfaces -- Exercises -- 6 Elliptic functions and integrals -- 6.1 Elliptic integrals -- 6.2 The Weierstrass [Omitted] function -- 6.3 Further topics -- Exercises -- 7 Applications of the Euler characteristic -- 7.1 The Euler characteristic and meromorphic forms -- 7.2 Applications -- Exercises -- PART III: DEEPER THEORY -- 8 Meromorphic functions and the Main Theorem for compact Riemann surfaces -- 8.1 Consequences of the Main Theorem -- 8.2 The Riemann-Roch formula -- Exercises -- 9 Proof of the Main Theorem -- 9.1 Discussion and motivation -- 9.2 The Riesz Representation Theorem -- 9.3 The heart of the proof -- 9.4 Weyl's Lemma -- Exercises -- 10 The Uniformisation Theorem -- 10.1 Statement -- 10.2 Proof of the analogue of the Main Theorem -- Exercises -- PART IV: FURTHER DEVELOPMENTS -- 11 Contrasts in Riemann surface theory -- 11.1 Algebraic aspects -- 11.2 Hyperbolic surfaces -- Exercises -- 12 Divisors, line bundles and Jacobians -- 12.1 Cohomology and line bundles -- 12.2 Jacobians of Riemann surfaces -- Exercises -- 13 Moduli and deformations -- 13.1 Almost-complex structures, Beltrami differentials and the integrability theorem -- 13.2 Deformations and cohomology -- 13.3 Appendix -- Exercises. 327 $a14 Mappings and moduli -- 14.1 Diffeomorphisms of the plane -- 14.2 Braids, Dehn twists and quadratic singularities -- 14.3 Hyperbolic geometry -- 14.4 Compactification of the moduli space -- Exercises -- 15 Ordinary differential equations -- 15.1 Conformal mapping -- 15.2 Periods of holomorphic forms and ordinary differential equations -- Exercises -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W. 330 $aAn authoritative but accessible text on one dimensional complex manifolds or Riemann surfaces. Dealing with the main results on Riemann surfaces from a variety of points of view; it pulls together material from global analysis, topology, and algebraic geometry, and covers the essential mathematical methods and tools. 606 $aRiemann surfaces 606 $aFunctions 615 0$aRiemann surfaces. 615 0$aFunctions. 676 $a515.93 676 $a515.93 700 $aDonaldson$b S. K$052791 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910811556803321 996 $aRiemann surfaces$93977334 997 $aUNINA