Geometry and Complexity Theory
About this book
Two central problems in computer science are P vs NP and the complexity of matrix multiplication. The first is also a leading candidate for the greatest unsolved problem in mathematics. The second is of enormous practical and theoretical importance. Algebraic geometry and representation theory provide fertile ground for advancing work on these problems and others in complexity. This introduction to algebraic complexity theory for graduate students and researchers in computer science and mathematics features concrete examples that demonstrate the application of geometric techniques to real world problems. Written by a noted expert in the field, it offers numerous open questions to motivate future research. Complexity theory has rejuvenated classical geometric questions and brought different areas of mathematics together in new ways. This book will show the beautiful, interesting, and important questions that have arisen as a result.
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Information
Table of contents
- Cover
- Half title
- Series
- Title
- Copyright
- Contents
- Preface
- 1 Introduction
- 2 The Complexity of Matrix Multiplication I: First Lower Bounds
- 3 The Complexity of Matrix Multiplication II: Asymptotic Upper Bounds
- 4 The Complexity of Matrix Multiplication III: Explicit Decompositions via Geometry
- 5 The Complexity of Matrix Multiplication IV: The Complexity of Tensors and More Lower Bounds
- 6 Valiant's Hypothesis I: Permanent versus Determinant and the Complexity of Polynomials
- 7 Valiant's Hypothesis II: Restricted Models and Other Approaches
- 8 Representation Theory and Its Uses in Complexity Theory
- 9 The Chow Variety of Products of Linear Forms
- 10 Topics Using Additional Algebraic Geometry
- Hints and Answers to Selected Exercises
- Bibliography
- Index
