Geometry and complexity theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom
Cambridge University Press
[2017]
|
| Schriftenreihe: | Cambridge studies in advanced mathematics
169 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xi, 339 Seiten Illustrationen, Diagramme |
| ISBN: | 9781107199231 |
Internformat
MARC
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Datensatz im Suchindex
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|---|---|
| adam_text | Titel: Geometry and complexity theory
Autor: Landsberg, Joseph M
Jahr: 2017
Contents
Preface page ix
1 Introduction 1
1.1 Matrix Multiplication 1
1.2 Separation of Algebraic Complexity Classes 11
1.3 How to Find Hay in a Haystack: The Problem
of Explicitness 17
1.4 The Role of Algebraic Geometry 17
2 The Complexity of Matrix Multiplication I:
First Lower Bounds 18
2.1 Matrix Multiplication and Multilinear Algebra 18
2.2 Strassen s Equations 25
2.3 Theory Needed for the Generalization of
Strassen s Equations 28
2.4 Koszul Flattenings 31
2.5 Matrix Multiplication and Koszul Flattenings 35
2.6 Lower Bounds for the Rank of Matrix Multiplication 42
3 The Complexity of Matrix Multiplication II:
Asymptotic Upper Bounds 45
3.1 Facts and Definitions from Algebraic Geometry 47
3.2 The Upper Bounds of Bini, Capovani, Lotti, and
Romani 54
3.3 Schönhage s Upper Bounds 55
3.4 Strassen s Laser Method 60
3.5 The Cohn-Umans Program 70
v
vi
Contents
4 The Complexity of Matrix Multiplication III:
Explicit Decompositions via Geometry 77
4.1 Symmetry and Decompositions 78
4.2 Two Decomposition Families of M(„) of Rank n3 81
4.3 Strassen s Decomposition Revisited 84
4.4 Invariants Associated to a Decomposition of M{n) 87
4.5 Cyclic Z3-Invariant Rank 23 Decompositions of M($) 90
4.6 Alternating Least Squares (ALS) Method for
Decompositions 93
4.7 Secant Varieties and Additional Geometrie Language 94
4.8 Border Rank Decompositions 98
5 The Complexity of Matrix Multiplication IV:
The Complexity of Tensors and More Lower Bounds 104
5.1 Tensors and Classical Linear Algebra 105
5.2 Indirectly Defined Equations 111
5.3 The Substitution Method 115
5.4 The Border Substitution Method 120
5.5 Geometry of the Coppersmith-Winograd Tensors 130
5.6 Ranks and Border Ranks of Structure Tensors of
Algebras 135
6 Valiant s Hypothesis I: Permanent versus Determinant
and the Complexity of Polynomials 144
6.1 Circuits and Definitions of VP and VNP 146
6.2 Flattenings: Our First Polynomials on the
Space of Polynomials 153
6.3 Singular Loci and Jacobian Varieties 157
6.4 Geometry and the State of the Art Regarding
dc(permm) 163
6.5 Extension of the Mignon-Ressayre Result to de 170
6.6 Symmetries of the Determinant and Permanent 173
6.7 de versus de 178
6.8 Determinantal Hypersurfaces 180
7 Valiant s Hypothesis H: Restricted Models and Other
Approaches 183
7.1 Waring Rank, Depth Three Powering Circuits,
and Symmetrie Polynomials 184
7.2 Depth Three Circuits and Secant Varieties of the
Chow Variety 188
Contents vii
7.3 Algebraic Branching Programs 191
7.4 Additional Restricted Models 197
7.5 Shallow Ciicuits and Valiant s Hypothesis 205
7.6 Hilbert Functions of Jacobian Ideals (Shifted Partial
Derivatives) and VP versus VNP 209
7.7 Polynomial Identity Testing, Hitting Sets, and Explicit
Noether Normalization 216
8 Representation Theory and Its Uses in Complexity Theory 221
8.1 Representation Theory of the General Linear Group 222
8.2 Young Flattenings 227
8.3 Additional Uses of Representation Theory to
Find Modules of Equations 230
8.4 Necessary Conditions for Modules of Polynomials
to Be Useful for GCT 233
8.5 Representation Theory and Vetn 235
8.6 Double-Commutant and Algebraic Peter-Wey 1
Theorems 238
8.7 Representations of Gj and GL(V) 245
8.8 The Program of [MS01, MS08] 248
8.9 Plethysm Coefficients, Kronecker Coefficients,
and Geometry 251
8.10 Orbit Occurrence Obstructions Cannot Separate
Perm™ from Vetn 253
8.11 Equivariant Determinantal Complexity 255
8.12 Symmetries of Additional Polynomials Relevant for
Complexity Theory 258
9 The Chow Variety of Products of Linear Forms 264
9.1 The Hermite-Hadamard-Howe Map 265
9.2 The GCT Perspective 269
9.3 6^-Formulation of the Hadamard-Howe Problem 271
9.4 Conjecture 9.1.2.18 and a Conjecture in Combinatorics 273
9.5 Algebraic Geometry Derivation of the
Hadamard-Howe Map 276
9.6 Brill s Equations 279
9.7 Proofs of Proposition 9.5.2.4 and Theorem 9.1.2.14 283
10 Topics Using Additional Algebraic Geometry 289
10.1 Rank and Cactus Rank of Polynomials 290
10.2 Cactus Varieties and Secant Varieties 297
viii
Contents
10.3 Nonnormality of T etn 299
10.4 The Minimal Free Resolution of the Ideal Generated by
Minors of Size r + 1 302
Hints and Answers to Selected Exercises 313
Bibliography 321
Index 335
|
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| discipline | Mathematik |
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| isbn | 9781107199231 |
| language | English |
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| series2 | Cambridge studies in advanced mathematics |
| spelling | Landsberg, Joseph M. 1963- (DE-588)14180677X aut Geometry and complexity theory J.M. Landsberg, Texas A&M University Cambridge, United Kingdom Cambridge University Press [2017] © 2017 xi, 339 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 169 Computational complexity Geometry, Algebraic Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Berechnungskomplexität (DE-588)4134751-1 gnd rswk-swf Berechnungskomplexität (DE-588)4134751-1 s Algebraische Geometrie (DE-588)4001161-6 s DE-604 Erscheint auch als Online-Ausgabe 978-1-108-18319-2 Cambridge studies in advanced mathematics 169 (DE-604)BV000003678 169 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029845308&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Landsberg, Joseph M. 1963- Geometry and complexity theory Cambridge studies in advanced mathematics Computational complexity Geometry, Algebraic Algebraische Geometrie (DE-588)4001161-6 gnd Berechnungskomplexität (DE-588)4134751-1 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4134751-1 |
| title | Geometry and complexity theory |
| title_auth | Geometry and complexity theory |
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| title_full | Geometry and complexity theory J.M. Landsberg, Texas A&M University |
| title_fullStr | Geometry and complexity theory J.M. Landsberg, Texas A&M University |
| title_full_unstemmed | Geometry and complexity theory J.M. Landsberg, Texas A&M University |
| title_short | Geometry and complexity theory |
| title_sort | geometry and complexity theory |
| topic | Computational complexity Geometry, Algebraic Algebraische Geometrie (DE-588)4001161-6 gnd Berechnungskomplexität (DE-588)4134751-1 gnd |
| topic_facet | Computational complexity Geometry, Algebraic Algebraische Geometrie Berechnungskomplexität |
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