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.

'The book under review provides an introduction accessible to graduate students and researchers either in computer science or in geometry and representation theory expanding a useful bridge between these disciplines and unifying recent trends.'

Felipe Zaldivar Source: MAA Reviews

'Most of the book is self-contained, since it introduces the necessary concepts of algebraic geometry and representation theory when needed. Several exercises are provided (for some of which hints and answers are available at the end of the book), together with an assessment of their difficulty and importance, stimulating the reader to an active reading.'

Matteo Gallet Source: MathSciNet

‘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 Source: Bulletin of the American Mathematical Society