The QGM Master Class Series 2012; 128 pp; softcover Volume: 2 ISBN10: 3037191082 ISBN13: 9783037191088 List Price: US$36 Member Price: US$28.80 Order Code: EMSQGM/2
 The term "categorification" was introduced by Louis Crane in 1995 and refers to the process of replacing settheoretic notions by the corresponding categorytheoretic analogues. This text mostly concentrates on algebraical aspects of the theory, presented in the historical perspective, but also contains several topological applications, in particular, an algebraic (or, more precisely, representationtheoretical) approach to categorification. It consists of fifteen sections corresponding to fifteen onehour lectures given during a Master Class at Aarhus University, Denmark in October 2010. There are some exercises collected at the end of the text and a rather extensive list of references. Video recordings of all (but one) lectures are available from the Master Class website. The book provides an introductory overview of the subject rather than a fully detailed monograph. The emphasis is made on definitions, examples and formulations of the results. Most proofs are either briefly outlined or omitted. However, complete proofs can be found by tracking references. It is assumed that the reader is familiar with the basics of category theory, representation theory, topology, and Lie algebra. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in algebra and algebraic geometry. Table of Contents  Basics: decategorification and categorification
 Basics: from categorification of linear maps to \(2\)categories
 Basics: \(2\)representations of finitary \(2\)categories
 Category \(\mathcal{O}\): definitions
 Category \(\mathcal{O}\): projective and shuffling functors
 Category \(\mathcal{O}\): twisting and completion
 Category \(\mathcal{O}\): grading and combinatorics
 \(\mathbb{S}_n\)categorification: Soergel bimodules, cells and Specht modules
 \(\mathbb{S}_n\)categorification: (induced) cell modules
 Category \(\mathcal{O}\): Koszul duality
 \(\mathfrak{sl}_2\)categorification: simple finitedimensional modules
 Application: categorification of the Jones polynomial
 \(\mathfrak{sl}_2\)categorification of Chuang and Rouquier
 Application: blocks of \(\mathbb{F}[\mathbb{S}_n]\) and Broué's conjecture
 Applications of \(\mathbb{S}_n\)categorifications
 Exercises
 Bibliography
 Index
