Holdings: Noncommutative geometry

Noncommutative geometry: a functorial approach

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Bibliographic Details
Main Author:	Nikolaev, Igor 1961- (Author)
Format:	Book
Language:	English
Published:	Berlin ; Boston De Gruyter [2022]
Edition:	2nd edition
Series:	De Gruyter Studies in Mathematics Volume 66
Subjects:	Nichtkommutative Geometrie Algebraische Geometrie Funktor Indextheorie Zahlentheorie Algebraic geometry, number theory, continuous geometries, topology, noncommutative algebraic geometry, quantum Arithmetic.
Online Access:	Inhaltsverzeichnis Inhaltsverzeichnis
Item Description:	Foreword to the second edition: "...contains two new chapters on the Arithmetic Topology and Quantum Arithmetic. All other parts remain intact."
Physical Description:	XIX, 378 Seiten Illustrationen 24 cm x 17 cm, 793 g
ISBN:	9783110788600 3110788608

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adam_text	CONTENTS FOREWORD - VII FOREWORD TO THE SECOND EDITION - IX INTRODUCTION ---- XI PART I: BASICS 1 1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.3 1.4 1.4.1 1.4.2 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 MODEL EXAMPLES - 3 NONCOMMUTATIVE TORI --- 3 GEOMETRIC DEFINITION ---- 3 ANALYTIC DEFINITION ---- 4 ALGEBRAIC DEFINITION ---- 5 ABSTRACT NONCOMMUTATIVE TORUS ---- 5 FIRST PROPERTIES OF THE AG ---- 6 ELLIPTIC CURVES ---- 7 WEIERSTRASS AND JACOBI NORMAL FORMS ---- 7 COMPLEX TORI ---- 8 WEIERSTRASS UNIFORMIZATION ---- 9 COMPLEX MULTIPLICATION ---- 10 FUNCTOR & - A E ---- 10 RANKS OF ELLIPTIC CURVES ---- 13 SYMMETRY OF COMPLEX AND REAL MULTIPLICATION ---- 14 ARITHMETIC COMPLEXITY OF THE ARM ---- 14 CLASSIFICATION OF SURFACE AUTOMORPHISMS ---- 15 ANOSOV AUTOMORPHISMS ---- 16 FUNCTOR F ---- 16 HANDELMAN S INVARIANT ---- 17 MODULE DETERMINANT ---- 18 NUMERICAL EXAMPLES ---- 18 2 2.1 2.2 2.3 CATEGORIES AND FUNCTORS - 23 CATEGORIES ---- 23 FUNCTORS ---- 26 NATURAL TRANSFORMATIONS ---- 28 XIV - CONTENTS 3 3.1 3.2 3.3 3.4 3.4.1 3.4.2 3.5 3.5.1 3.5.2 3.6 3.7 C-ALGEBRAS - 31 BASIC DEFINITIONS ---- 31 CROSSED PRODUCTS ---- 33 K-THEORY OFTHE C-ALGEBRAS ---- 35 NONCOMMUTATIVE TORI ---- 39 N-DIMENSIONAL NONCOMMUTATIVE TORI ---- 39 2-DIMENSIONAL NONCOMMUTATIVE TORI ---- 41 AF-ALGEBRAS ---- 43 GENERIC AF-ALGEBRAS ---- 43 STATIONARY AF-ALGEBRAS ---- 46 UHF-ALGEBRAS ---- 47 CUNTZ-KRIEGER ALGEBRAS ---- 49 PART II: NONCOMMUTATIVE INVARIANTS 4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 TOPOLOGY ---- 55 CLASSIFICATION OFTHE SURFACE AUTOMORPHISMS ---- 55 PSEUDO-ANOSOV AUTOMORPHISMS OF A SURFACE ---- 55 FUNCTORS AND INVARIANTS ---- 57 JACOBIAN OF MEASURED FOLIATIONS ---- 59 EQUIVALENT FOLIATIONS ---- 61 PROOFS ---- 62 ANOSOV MAPS OFTHE TORUS ---- 67 TORSION IN THE TORUS BUNDLES ---- 70 CUNTZ-KRIEGER FUNCTOR ---- 70 PROOF OF THEOREM 4.2.1 ---- 71 NONCOMMUTATIVE INVARIANTS OF TORUS BUNDLES ---- 73 OBSTRUCTION THEORY FOR ANOSOV S BUNDLES ---- 75 FUNDAMENTAL AF-ALGEBRA ---- 75 PROOFS ---- 78 OBSTRUCTION THEORY ---- 83 CLUSTER C-ALGEBRAS AND KNOT POLYNOMIALS -----86 INVARIANT LAURENT POLYNOMIALS -----86 BIRMAN-HILDEN THEOREM ---- 88 CLUSTER C -ALGEBRAS -----89 JONES AND HOMFLY POLYNOMIALS ---- 91 PROOF OF THEOREM 4.4.1 ---- 92 EXAMPLES ---- 97 5 5.1 ALGEBRAIC GEOMETRY - 103 ELLIPTIC CURVES ---- 103 CONTENTS - - XV 5.1.1 NONCOMMUTATIVE TORI VIA SKLYANIN ALGEBRAS ---- 104 5.1.2 NONCOMMUTATIVE TORI VIA MEASURED FOLIATIONS ---- 109 5.2 ALGEBRAIC CURVES OF GENUS G 1 ----- 114 5.2.1 TORIC AF-ALGEBRAS ---- 115 5.2.2 PROOF OF THEOREM 5.2.1 ---- 116 5.3 PROJECTIVE VARIETIES OF DIMENSION N 1 ---- 120 5.3.1 SERRE C-ALGEBRAS ---- 121 5.3.2 PROOF OF THEOREM 5.3.4 ---- 124 5.3.3 EXAMPLE ---- 127 5.4 TATE CURVES AND UHF-ALGEBRAS ---- 129 5.4.1 ELLIPTIC CURVE OVER P-ADIC NUMBERS ---- 129 5.4.2 PROOF OF THEOREM 5.4.1 ----- 130 5.4.3 EXAMPLE ----- 134 5.5 MAPPING CLASS GROUP ---- 135 5.5.1 HARVEY S CONJECTURE ---- 135 5.5.2 PROOF OF THEOREM 5.5.1 ----- 136 6 NUMBER THEORY ----- 143 6.1 ISOGENIES OF ELLIPTIC CURVES ---- 143 6.1.1 SYMMETRY OF COMPLEX AND REAL MULTIPLICATION ---- 144 6.1.2 PROOF OF THEOREM 6.1.1 ----- 145 6.1.3 PROOF OF THEOREM 6.1.2 ----- 148 6.1.4 PROOF OF THEOREM 6.1.3 ----- 150 6.2 RANKS OF ELLIPTIC CURVES ----- 155 6.2.1 ARITHMETIC COMPLEXITY OF ARM ---- 155 6.2.2 MORDELL AF-ALGEBRA ----- 156 6.2.3 PROOF OF THEOREM 6.2.1 ----- 157 6.2.4 NUMERICAL EXAMPLES ----- 161 6.3 TRANSCENDENTAL NUMBER THEORY ---- 162 6.3.1 ALGEBRAIC VALUES OF Y(0,) = E2ME+LOGLOGE - - - - 162 6.3.2 PROOF OF THEOREM 6.3.1 ----- 163 6.3.3 COMMENTS ON A NOTE BY M. WALDSCHMIDT ---- 165 6.4 CLASS FIELD THEORY ---- 167 6.4.1 HILBERT CLASS FIELD OF A REAL QUADRATIC FIELD ---- 167 6.4.2 AF-ALGEBRA OFTHE HECKE EIGENFORM ----- 169 6.4.3 PROOF OF THEOREM 6.4.1 ----- 170 6.4.4 EXAMPLES ----- 173 6.5 NONCOMMUTATIVE RECIPROCITY ----- 175 6.5.1 /.-FUNCTION OF NONCOMMUTATIVE TORI ----- 175 6.5.2 PROOF OF THEOREM 6.5.1 ----- 176 6.5.3 SUPPLEMENT: GRBSSENCHARACTERS, UNITS, AND N(N) ----- 182 6.6 LANGLANDS CONJECTURE FOR AZM ----- 185 XVI ----- CONTENTS 6.6.1 6.6.2 6.6.3 6.7 6.7.1 6.7.2 6.7.3 L(ARM,S) ----- 185 PROOF OF THEOREM 6.6.1 ---- 189 SUPPLEMENT: ARTIN /.-FUNCTION ---- 191 PROJECTIVE VARIETIES OVER FINITE FIELDS ---- 193 TRACES OF FROBENIUS ENDOMORPHISMS ---- 193 PROOF OF THEOREM 6.7.1 ---- 195 EXAMPLES ---- 200 7 7.1 7.1.1 7.1.2 7.1.3 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.3 7.3.1 7.3.2 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.5 7.5.1 7.5.2 7.5.3 ARITHMETIC TOPOLOGY ---- 207 ARITHMETIC TOPOLOGY OF 3-MANIFOLDS ---- 207 BRAIDS, LINKS, AND GALOIS COVERING ---- 209 PROOF OF THEOREM 7.1.1 ---- 210 PUNCTURED TORUS ---- 213 ARITHMETIC TOPOLOGY OF 4-MANIFOLDS ---- 215 4-DIMENSIONAL MANIFOLDS ---- 216 GALOIS THEORY FOR NONCOMMUTATIVE FIELDS ---- 217 UCHIDA MAP ---- 218 PROOFS ---- 219 ROKHLIN AND DONALDSON S THEOREMS REVISITED ---- 222 UNTYING KNOTS IN 4D AND WEDDERBURN S THEOREM ---- 225 WEDDERBURN S THEOREM ---- 225 PROOF OF THEOREM 7.3.1 ---- 226 DYNAMICAL IDEALS OF NONCOMMUTATIVE RINGS ---- 227 TOPOLOGICAL DYNAMICS - 230 CYCLIC DIVISION ALGEBRAS ---- 230 PIERGALLINI COVERING ---- 231 PROOF OF THEOREM 7.4.1 ---- 232 KNOTTED SURFACES IN 4-MANIFOLDS ---- 234 ETESI C -ALGEBRAS ---- 238 MINKOWSKI GROUP ---- 239 GOMPF S THEOREM ---- 240 PROOFS ---- 241 8 8.1 8.1.1 8.1.2 8.1.3 8.1.4 8.2 8.2.1 8.2.2 QUANTUM ARITHMETIC ---- 249 LANGLANDS RECIPROCITY FOR C-ALGEBRAS ---- 249 TRACE COHOMOLOGY ---- 250 LANGLANDS RECIPROCITY ---- 251 PROOFS ---- 252 PIMSNER-VOICULESCU EMBEDDING ---- 257 K-THEORY OF RATIONAL QUADRATIC FORMS ---- 259 ALGEBRAIC GROUPS OVER ADELES ---- 260 PROOFS ---- 261 CONTENTS XVII 8.2.3 8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 8.5 8.5.1 8.5.2 8.5.3 8.5.4 8.6 8.6.1 8.6.2 BINARY QUADRATIC FORMS ---- 264 QUANTUM DYNAMICS OF ELLIPTIC CURVES ---- 265 C-DYNAMICAL SYSTEMS ---- 267 ABELIAN EXTENSIONS OF QUADRATIC FIELDS ---- 267 SHAFAREVICH-TATE GROUP OF ELLIPTIC CURVES ---- 268 PROOFS ---- 268 SHAFAREVICH-TATE GROUPS OF ABELIAN VARIETIES ---- 275 ABELIAN VARIETIES ---- 276 WEIL-CHATELET GROUP ---- 277 LOCALIZATION FORMULAS ---- 278 PROOF OF THEOREM 8.4.1 ---- 279 ABELIAN VARIETIES WITH COMPLEX MULTIPLICATION ---- 283 NONCOMMUTATIVE GEOMETRY OF ELLIPTIC SURFACES ---- 285 BROCK-ELKIES-JORDAN VARIETY ---- 286 ELLIPTIC SURFACES ---- 288 PROOFS ---- 288 PICARD NUMBERS ---- 294 CLASS FIELD TOWERS AND MINIMAL MODELS ---- 295 ALGEBRAIC SURFACES ---- 297 PROOFS ---- 297 PART III: BRIEF SURVEY OF NCG 9 9.1 9.2 9.3 FINITE GEOMETRIES ---- 305 AXIOMS OF PROJECTIVE GEOMETRY ---- 305 PROJECTIVE SPACES OVER SKEW FIELDS ---- 306 DESARGUES AND PAPPUS AXIOMS ---- 307 10 10.1 10.2 CONTINUOUS GEOMETRIES ---- 309 W* -ALGEBRAS ---- 309 VON NEUMANN GEOMETRY ---- 311 11 11.1 11.1.1 11.1.2 11.2 11.2.1 11.2.2 11.2.3 11.3 CONNES GEOMETRIES ---- 313 CLASSIFICATION OF TYPE III FACTORS ---- 313 TOMITA-TAKESAKI THEORY ---- 313 CONNES INVARIANTS ---- 314 NONCOMMUTATIVE DIFFERENTIAL GEOMETRY ---- 315 HOCHSCHILD HOMOLOGY ---- 315 CYCLIC HOMOLOGY ---- 316 NOVIKOV CONJECTURE FOR HYPERBOLIC GROUPS ---- 317 CONNES INDEX THEOREM ---- 318 XVIII - CONTENTS 11.3.1 11.3.2 11.3.3 11.4 11.4.1 11.4.2 ATIYAH-SINGER THEOREM FOR FAMILIES OF ELLIPTIC OPERATORS ---- 318 FOLIATED SPACES ---- 319 INDEX THEOREM FOR FOLIATED SPACES ---- 320 BOST-CONNES DYNAMICAL SYSTEM ---- 321 HECKE C*-ALGEBRA ---- 321 BOST-CONNES THEOREM ---- 322 12 12.1 12.1.1 12.1.2 12.1.3 12.2 12.2.1 12.2.2 12.2.3 12.3 12.3.1 12.3.2 12.4 12.4.1 12.4.2 12.4.3 12.5 INDEX THEORY ---- 325 ATIYAH-SINGER THEOREM ---- 325 FREDHOLM OPERATORS ---- 325 ELLIPTIC OPERATORS ON MANIFOLDS ---- 326 INDEX THEOREM ---- 327 K-HOMOLOGY ---- 328 TOPOLOGICAL K-THEORY ---- 328 ATIAYH S REALIZATION OF K-HOMOLOGY ---- 330 BROWN-DOUGLAS-FILLMORE THEORY ---- 331 KASPAROV S KK-THEORY ---- 332 HILBERT MODULES ---- 333 KK-GROUPS ---- 334 APPLICATIONS OF INDEX THEORY ---- 335 NOVIKOV CONJECTURE ---- 335 BAUM-CONNES CONJECTURE ---- 337 POSITIVE SCALAR CURVATURE ---- 337 COARSE GEOMETRY ---- 338 13 13.1 13.2 13.3 JONES POLYNOMIALS - 341 SUBFACTORS ---- 341 BRAIDS ---- 342 TRACE INVARIANT ---- 344 14 14.1 14.2 14.3 QUANTUM GROUPS---- 347 MANIN S QUANTUM PLANE ---- 348 HOPF ALGEBRAS ---- 349 OPERATOR ALGEBRAS AND QUANTUM GROUPS ---- 350 15 15.1 15.2 15.3 NONCOMMUTATIVE ALGEBRAIC GEOMETRY - 353 SERRE ISOMORPHISM ---- 353 TWISTED HOMOGENEOUS COORDINATE RINGS ---- 355 SKLYANIN ALGEBRAS ---- 356 16 16.1 TRENDS IN NONCOMMUTATIVE GEOMETRY - 359 DERIVED CATEGORIES ---- 359 CONTENTS - - XIX 16.2 NONCOMMUTATIVE THICKENING ----- 360 16.3 DEFORMATION QUANTIZATION OF POISSON MANIFOLDS ----- 361 16.4 ALGEBRAIC GEOMETRY OF NONCOMMUTATIVE RINGS ---- 362 BIBLIOGRAPHY ---- 365 INDEX ---- 375
adam_txt	CONTENTS FOREWORD - VII FOREWORD TO THE SECOND EDITION - IX INTRODUCTION ---- XI PART I: BASICS 1 1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.3 1.4 1.4.1 1.4.2 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 MODEL EXAMPLES - 3 NONCOMMUTATIVE TORI --- 3 GEOMETRIC DEFINITION ---- 3 ANALYTIC DEFINITION ---- 4 ALGEBRAIC DEFINITION ---- 5 ABSTRACT NONCOMMUTATIVE TORUS ---- 5 FIRST PROPERTIES OF THE AG ---- 6 ELLIPTIC CURVES ---- 7 WEIERSTRASS AND JACOBI NORMAL FORMS ---- 7 COMPLEX TORI ---- 8 WEIERSTRASS UNIFORMIZATION ---- 9 COMPLEX MULTIPLICATION ---- 10 FUNCTOR & - A E ---- 10 RANKS OF ELLIPTIC CURVES ---- 13 SYMMETRY OF COMPLEX AND REAL MULTIPLICATION ---- 14 ARITHMETIC COMPLEXITY OF THE ARM ---- 14 CLASSIFICATION OF SURFACE AUTOMORPHISMS ---- 15 ANOSOV AUTOMORPHISMS ---- 16 FUNCTOR F ---- 16 HANDELMAN ' S INVARIANT ---- 17 MODULE DETERMINANT ---- 18 NUMERICAL EXAMPLES ---- 18 2 2.1 2.2 2.3 CATEGORIES AND FUNCTORS - 23 CATEGORIES ---- 23 FUNCTORS ---- 26 NATURAL TRANSFORMATIONS ---- 28 XIV - CONTENTS 3 3.1 3.2 3.3 3.4 3.4.1 3.4.2 3.5 3.5.1 3.5.2 3.6 3.7 C-ALGEBRAS - 31 BASIC DEFINITIONS ---- 31 CROSSED PRODUCTS ---- 33 K-THEORY OFTHE C-ALGEBRAS ---- 35 NONCOMMUTATIVE TORI ---- 39 N-DIMENSIONAL NONCOMMUTATIVE TORI ---- 39 2-DIMENSIONAL NONCOMMUTATIVE TORI ---- 41 AF-ALGEBRAS ---- 43 GENERIC AF-ALGEBRAS ---- 43 STATIONARY AF-ALGEBRAS ---- 46 UHF-ALGEBRAS ---- 47 CUNTZ-KRIEGER ALGEBRAS ---- 49 PART II: NONCOMMUTATIVE INVARIANTS 4 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 TOPOLOGY ---- 55 CLASSIFICATION OFTHE SURFACE AUTOMORPHISMS ---- 55 PSEUDO-ANOSOV AUTOMORPHISMS OF A SURFACE ---- 55 FUNCTORS AND INVARIANTS ---- 57 JACOBIAN OF MEASURED FOLIATIONS ---- 59 EQUIVALENT FOLIATIONS ---- 61 PROOFS ---- 62 ANOSOV MAPS OFTHE TORUS ---- 67 TORSION IN THE TORUS BUNDLES ---- 70 CUNTZ-KRIEGER FUNCTOR ---- 70 PROOF OF THEOREM 4.2.1 ---- 71 NONCOMMUTATIVE INVARIANTS OF TORUS BUNDLES ---- 73 OBSTRUCTION THEORY FOR ANOSOV ' S BUNDLES ---- 75 FUNDAMENTAL AF-ALGEBRA ---- 75 PROOFS ---- 78 OBSTRUCTION THEORY ---- 83 CLUSTER C-ALGEBRAS AND KNOT POLYNOMIALS -----86 INVARIANT LAURENT POLYNOMIALS -----86 BIRMAN-HILDEN THEOREM ---- 88 CLUSTER C -ALGEBRAS -----89 JONES AND HOMFLY POLYNOMIALS ---- 91 PROOF OF THEOREM 4.4.1 ---- 92 EXAMPLES ---- 97 5 5.1 ALGEBRAIC GEOMETRY - 103 ELLIPTIC CURVES ---- 103 CONTENTS - - XV 5.1.1 NONCOMMUTATIVE TORI VIA SKLYANIN ALGEBRAS ---- 104 5.1.2 NONCOMMUTATIVE TORI VIA MEASURED FOLIATIONS ---- 109 5.2 ALGEBRAIC CURVES OF GENUS G 1 ----- 114 5.2.1 TORIC AF-ALGEBRAS ---- 115 5.2.2 PROOF OF THEOREM 5.2.1 ---- 116 5.3 PROJECTIVE VARIETIES OF DIMENSION N 1 ---- 120 5.3.1 SERRE C-ALGEBRAS ---- 121 5.3.2 PROOF OF THEOREM 5.3.4 ---- 124 5.3.3 EXAMPLE ---- 127 5.4 TATE CURVES AND UHF-ALGEBRAS ---- 129 5.4.1 ELLIPTIC CURVE OVER P-ADIC NUMBERS ---- 129 5.4.2 PROOF OF THEOREM 5.4.1 ----- 130 5.4.3 EXAMPLE ----- 134 5.5 MAPPING CLASS GROUP ---- 135 5.5.1 HARVEY ' S CONJECTURE ---- 135 5.5.2 PROOF OF THEOREM 5.5.1 ----- 136 6 NUMBER THEORY ----- 143 6.1 ISOGENIES OF ELLIPTIC CURVES ---- 143 6.1.1 SYMMETRY OF COMPLEX AND REAL MULTIPLICATION ---- 144 6.1.2 PROOF OF THEOREM 6.1.1 ----- 145 6.1.3 PROOF OF THEOREM 6.1.2 ----- 148 6.1.4 PROOF OF THEOREM 6.1.3 ----- 150 6.2 RANKS OF ELLIPTIC CURVES ----- 155 6.2.1 ARITHMETIC COMPLEXITY OF ARM ---- 155 6.2.2 MORDELL AF-ALGEBRA ----- 156 6.2.3 PROOF OF THEOREM 6.2.1 ----- 157 6.2.4 NUMERICAL EXAMPLES ----- 161 6.3 TRANSCENDENTAL NUMBER THEORY ---- 162 6.3.1 ALGEBRAIC VALUES OF Y(0,) = E2ME+LOGLOGE - - - - 162 6.3.2 PROOF OF THEOREM 6.3.1 ----- 163 6.3.3 COMMENTS ON A NOTE BY M. WALDSCHMIDT ---- 165 6.4 CLASS FIELD THEORY ---- 167 6.4.1 HILBERT CLASS FIELD OF A REAL QUADRATIC FIELD ---- 167 6.4.2 AF-ALGEBRA OFTHE HECKE EIGENFORM ----- 169 6.4.3 PROOF OF THEOREM 6.4.1 ----- 170 6.4.4 EXAMPLES ----- 173 6.5 NONCOMMUTATIVE RECIPROCITY ----- 175 6.5.1 /.-FUNCTION OF NONCOMMUTATIVE TORI ----- 175 6.5.2 PROOF OF THEOREM 6.5.1 ----- 176 6.5.3 SUPPLEMENT: GRBSSENCHARACTERS, UNITS, AND N(N) ----- 182 6.6 LANGLANDS CONJECTURE FOR AZM ----- 185 XVI ----- CONTENTS 6.6.1 6.6.2 6.6.3 6.7 6.7.1 6.7.2 6.7.3 L(ARM,S) ----- 185 PROOF OF THEOREM 6.6.1 ---- 189 SUPPLEMENT: ARTIN /.-FUNCTION ---- 191 PROJECTIVE VARIETIES OVER FINITE FIELDS ---- 193 TRACES OF FROBENIUS ENDOMORPHISMS ---- 193 PROOF OF THEOREM 6.7.1 ---- 195 EXAMPLES ---- 200 7 7.1 7.1.1 7.1.2 7.1.3 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.3 7.3.1 7.3.2 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5 7.5 7.5.1 7.5.2 7.5.3 ARITHMETIC TOPOLOGY ---- 207 ARITHMETIC TOPOLOGY OF 3-MANIFOLDS ---- 207 BRAIDS, LINKS, AND GALOIS COVERING ---- 209 PROOF OF THEOREM 7.1.1 ---- 210 PUNCTURED TORUS ---- 213 ARITHMETIC TOPOLOGY OF 4-MANIFOLDS ---- 215 4-DIMENSIONAL MANIFOLDS ---- 216 GALOIS THEORY FOR NONCOMMUTATIVE FIELDS ---- 217 UCHIDA MAP ---- 218 PROOFS ---- 219 ROKHLIN AND DONALDSON ' S THEOREMS REVISITED ---- 222 UNTYING KNOTS IN 4D AND WEDDERBURN ' S THEOREM ---- 225 WEDDERBURN ' S THEOREM ---- 225 PROOF OF THEOREM 7.3.1 ---- 226 DYNAMICAL IDEALS OF NONCOMMUTATIVE RINGS ---- 227 TOPOLOGICAL DYNAMICS - 230 CYCLIC DIVISION ALGEBRAS ---- 230 PIERGALLINI COVERING ---- 231 PROOF OF THEOREM 7.4.1 ---- 232 KNOTTED SURFACES IN 4-MANIFOLDS ---- 234 ETESI C -ALGEBRAS ---- 238 MINKOWSKI GROUP ---- 239 GOMPF ' S THEOREM ---- 240 PROOFS ---- 241 8 8.1 8.1.1 8.1.2 8.1.3 8.1.4 8.2 8.2.1 8.2.2 QUANTUM ARITHMETIC ---- 249 LANGLANDS RECIPROCITY FOR C-ALGEBRAS ---- 249 TRACE COHOMOLOGY ---- 250 LANGLANDS RECIPROCITY ---- 251 PROOFS ---- 252 PIMSNER-VOICULESCU EMBEDDING ---- 257 K-THEORY OF RATIONAL QUADRATIC FORMS ---- 259 ALGEBRAIC GROUPS OVER ADELES ---- 260 PROOFS ---- 261 CONTENTS XVII 8.2.3 8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 8.5 8.5.1 8.5.2 8.5.3 8.5.4 8.6 8.6.1 8.6.2 BINARY QUADRATIC FORMS ---- 264 QUANTUM DYNAMICS OF ELLIPTIC CURVES ---- 265 C-DYNAMICAL SYSTEMS ---- 267 ABELIAN EXTENSIONS OF QUADRATIC FIELDS ---- 267 SHAFAREVICH-TATE GROUP OF ELLIPTIC CURVES ---- 268 PROOFS ---- 268 SHAFAREVICH-TATE GROUPS OF ABELIAN VARIETIES ---- 275 ABELIAN VARIETIES ---- 276 WEIL-CHATELET GROUP ---- 277 LOCALIZATION FORMULAS ---- 278 PROOF OF THEOREM 8.4.1 ---- 279 ABELIAN VARIETIES WITH COMPLEX MULTIPLICATION ---- 283 NONCOMMUTATIVE GEOMETRY OF ELLIPTIC SURFACES ---- 285 BROCK-ELKIES-JORDAN VARIETY ---- 286 ELLIPTIC SURFACES ---- 288 PROOFS ---- 288 PICARD NUMBERS ---- 294 CLASS FIELD TOWERS AND MINIMAL MODELS ---- 295 ALGEBRAIC SURFACES ---- 297 PROOFS ---- 297 PART III: BRIEF SURVEY OF NCG 9 9.1 9.2 9.3 FINITE GEOMETRIES ---- 305 AXIOMS OF PROJECTIVE GEOMETRY ---- 305 PROJECTIVE SPACES OVER SKEW FIELDS ---- 306 DESARGUES AND PAPPUS AXIOMS ---- 307 10 10.1 10.2 CONTINUOUS GEOMETRIES ---- 309 W* -ALGEBRAS ---- 309 VON NEUMANN GEOMETRY ---- 311 11 11.1 11.1.1 11.1.2 11.2 11.2.1 11.2.2 11.2.3 11.3 CONNES GEOMETRIES ---- 313 CLASSIFICATION OF TYPE III FACTORS ---- 313 TOMITA-TAKESAKI THEORY ---- 313 CONNES INVARIANTS ---- 314 NONCOMMUTATIVE DIFFERENTIAL GEOMETRY ---- 315 HOCHSCHILD HOMOLOGY ---- 315 CYCLIC HOMOLOGY ---- 316 NOVIKOV CONJECTURE FOR HYPERBOLIC GROUPS ---- 317 CONNES ' INDEX THEOREM ---- 318 XVIII - CONTENTS 11.3.1 11.3.2 11.3.3 11.4 11.4.1 11.4.2 ATIYAH-SINGER THEOREM FOR FAMILIES OF ELLIPTIC OPERATORS ---- 318 FOLIATED SPACES ---- 319 INDEX THEOREM FOR FOLIATED SPACES ---- 320 BOST-CONNES DYNAMICAL SYSTEM ---- 321 HECKE C*-ALGEBRA ---- 321 BOST-CONNES THEOREM ---- 322 12 12.1 12.1.1 12.1.2 12.1.3 12.2 12.2.1 12.2.2 12.2.3 12.3 12.3.1 12.3.2 12.4 12.4.1 12.4.2 12.4.3 12.5 INDEX THEORY ---- 325 ATIYAH-SINGER THEOREM ---- 325 FREDHOLM OPERATORS ---- 325 ELLIPTIC OPERATORS ON MANIFOLDS ---- 326 INDEX THEOREM ---- 327 K-HOMOLOGY ---- 328 TOPOLOGICAL K-THEORY ---- 328 ATIAYH ' S REALIZATION OF K-HOMOLOGY ---- 330 BROWN-DOUGLAS-FILLMORE THEORY ---- 331 KASPAROV ' S KK-THEORY ---- 332 HILBERT MODULES ---- 333 KK-GROUPS ---- 334 APPLICATIONS OF INDEX THEORY ---- 335 NOVIKOV CONJECTURE ---- 335 BAUM-CONNES CONJECTURE ---- 337 POSITIVE SCALAR CURVATURE ---- 337 COARSE GEOMETRY ---- 338 13 13.1 13.2 13.3 JONES POLYNOMIALS - 341 SUBFACTORS ---- 341 BRAIDS ---- 342 TRACE INVARIANT ---- 344 14 14.1 14.2 14.3 QUANTUM GROUPS---- 347 MANIN ' S QUANTUM PLANE ---- 348 HOPF ALGEBRAS ---- 349 OPERATOR ALGEBRAS AND QUANTUM GROUPS ---- 350 15 15.1 15.2 15.3 NONCOMMUTATIVE ALGEBRAIC GEOMETRY - 353 SERRE ISOMORPHISM ---- 353 TWISTED HOMOGENEOUS COORDINATE RINGS ---- 355 SKLYANIN ALGEBRAS ---- 356 16 16.1 TRENDS IN NONCOMMUTATIVE GEOMETRY - 359 DERIVED CATEGORIES ---- 359 CONTENTS - - XIX 16.2 NONCOMMUTATIVE THICKENING ----- 360 16.3 DEFORMATION QUANTIZATION OF POISSON MANIFOLDS ----- 361 16.4 ALGEBRAIC GEOMETRY OF NONCOMMUTATIVE RINGS ---- 362 BIBLIOGRAPHY ---- 365 INDEX ---- 375
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spelling	Nikolaev, Igor 1961- (DE-588)12105358X aut Noncommutative geometry a functorial approach Igor V. Nikolaev 2nd edition Berlin ; Boston De Gruyter [2022] © 2022 XIX, 378 Seiten Illustrationen 24 cm x 17 cm, 793 g txt rdacontent n rdamedia nc rdacarrier De Gruyter Studies in Mathematics Volume 66 Foreword to the second edition: "...contains two new chapters on the Arithmetic Topology and Quantum Arithmetic. All other parts remain intact." Nichtkommutative Geometrie (DE-588)4171742-9 gnd rswk-swf Nichtkommutative Geometrie Algebraische Geometrie Funktor Indextheorie Zahlentheorie Algebraic geometry, number theory, continuous geometries, topology, noncommutative algebraic geometry, quantum Arithmetic. Nichtkommutative Geometrie (DE-588)4171742-9 s DE-604 Walter de Gruyter GmbH & Co. KG (DE-588)10095502-2 pbl Erscheint auch als Online-Ausgabe, EPUB 978-3-11-078881-5 Erscheint auch als Online-Ausgabe, PDF 978-3-11-078870-9 De Gruyter Studies in Mathematics Volume 66 (DE-604)BV000005407 66 B:DE-101 application/pdf https://d-nb.info/1253738122/04 Inhaltsverzeichnis DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033875562&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p vlb 20220319 DE-101 https://d-nb.info/provenance/plan#vlb
spellingShingle	Nikolaev, Igor 1961- Noncommutative geometry a functorial approach De Gruyter Studies in Mathematics Nichtkommutative Geometrie (DE-588)4171742-9 gnd
subject_GND	(DE-588)4171742-9
title	Noncommutative geometry a functorial approach
title_auth	Noncommutative geometry a functorial approach
title_exact_search	Noncommutative geometry a functorial approach
title_exact_search_txtP	Noncommutative geometry a functorial approach
title_full	Noncommutative geometry a functorial approach Igor V. Nikolaev
title_fullStr	Noncommutative geometry a functorial approach Igor V. Nikolaev
title_full_unstemmed	Noncommutative geometry a functorial approach Igor V. Nikolaev
title_short	Noncommutative geometry
title_sort	noncommutative geometry a functorial approach
title_sub	a functorial approach
topic	Nichtkommutative Geometrie (DE-588)4171742-9 gnd
topic_facet	Nichtkommutative Geometrie
url	https://d-nb.info/1253738122/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033875562&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000005407
work_keys_str_mv	AT nikolaevigor noncommutativegeometryafunctorialapproach AT walterdegruytergmbhcokg noncommutativegeometryafunctorialapproach

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