Geometry and Complexity Theory

This comprehensive introduction to algebraic complexity theory presents new techniques for analyzing P vs NP and matrix multiplication.

J. M. Landsberg (Author)

9781107199231, Cambridge University Press

Hardback, published 28 September 2017

350 pages
23.5 x 15.6 x 2.4 cm, 0.62 kg

'We greatly encourage mathematicians interested in these subjects (algebraic geometers in particular, but not only!) to find many, many more interesting results in the Geometry and Complexity Theory by J. M. Landsberg.' Mateusz Michalek, Bulletin of the American Mathematical Society

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.

1. Introduction
2. The complexity of matrix multiplication I
3. The complexity of matrix multiplication II
4. The complexity of matrix multiplication III
5. The complexity of matrix multiplication IV
6. Valiant's hypothesis I
7. Valiant's hypothesis II
8. Representation theory and its uses in complexity theory
9. The Chow variety of products of linear forms
10. Topics using additional algebraic geometry.

Subject Areas: Algorithms & data structures [UMB], Algebraic geometry [PBMW]