This book describes recent applications of algebraic geometry and representation theory to complexity theory. I focus on two central problems: the complexity of matrix multiplication and Valiant's algebraic variants ofPversusNP.

I have attempted to make this book accessible to both computer scientists and geometers and the exposition as self-contained as possible. Two goals are to convince computer scientists of the utility of techniques from algebraic geometry and representation theory and to show geometers beautiful, interesting, and important geometry questions arising in complexity theory.

Computer scientists have made extensive use combinatorics, graph theory, probability, and linear algebra. I hope to show that even elementary techniques from algebraic geometry and representation theory can substantially advance the search for lower bounds, and even upper bounds, in complexity theory. I believe such additional mathematics will be necessary for further advances on questions discussed in this book as well as related complexity problems. Techniques are introduced as needed to deal with concrete problems.

For geometers, I expect that complexity theory will be as good a source for questions in algebraic geometry as has been modern physics. Recent work has indicated that subjects such as Fulton-McPherson intersection theory, the Hilbert scheme of points, and the Kempf-Weyman method for computing syzygies all have something to add to complexity theory. In addition, complexity theory has a way of rejuvenating old questions that had been nearly forgotten but remain beautiful and intriguing: questions of Hadamard, Darboux, Lüroth, and the classical Italian school. At the same time, complexity theory has brought different areas of mathematics together in new ways: for instance, combinatorics, representation theory, and algebraic geometry all play a role in understanding the coordinate ring of the orbit closure of the determinant.

This book evolved from several classes I have given on the subject: a spring 2013 semester course at Texas A&M; summer courses at Scuola Matematica Inter-universitaria, Cortona (July 2012), CIRM, Trento (June 2014), the University of Chicago (IMA sponsored) (July 2014), KAIST, Deajeon (August 2015), and Obergurgul, Austria (September 2016); a fall 2016 semester course at Texas A&M; and, most importantly, a fall 2014 semester course at the University of California, Berkeley, as part of the semester-long program Algorithms and Complexity in Algebraic Geometry at the Simons Institute for the Theory of Computing.