Memoirs of the American Mathematical Society 2011; 86 pp; softcover Volume: 215 ISBN10: 0821853082 ISBN13: 9780821853085 List Price: US$70 Individual Members: US$42 Institutional Members: US$56 Order Code: MEMO/215/1011
 In this article the author uses techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3dimensional CalabiYau algebras. The CalabiYau property appears throughout geometry and string theory and is increasingly being studied in algebra. He further shows that the algebras constructed are examples of noncommutative crepant resolutions (NCCRs), in the sense of Van den Bergh, of Gorenstein affine toric threefolds. Dimer models, first studied in theoretical physics, give a way of writing down a class of noncommutative algebras, as the path algebra of a quiver with relations obtained from a `superpotential'. Some examples are CalabiYau and some are not. The author considers two types of `consistency' conditions on dimer models, and shows that a `geometrically consistent' dimer model is `algebraically consistent'. He proves that the algebras obtained from algebraically consistent dimer models are 3dimensional CalabiYau algebras. This is the key step which allows him to prove that these algebras are NCCRs of the Gorenstein affine toric threefolds associated to the dimer models. Table of Contents  Introduction
 Introduction to the dimer model
 Consistency
 Zigzag flows and perfect matchings
 Toric algebras and algebraic consistency
 Geometric consistency implies algebraic consistency
 CalabiYau algebras from algebraically consistent dimers
 Noncommutative crepant resolutions
 Bibliography
