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Klappentext In the 2012-13 academic year! the Mathematical Sciences Research Institute! Berkeley! hosted programs in Commutative Algebra (Fall 2012 and Spring 2013) and Noncommutative Algebraic Geometry and Representation Theory (Spring 2013). There have been many significant developments in these fields in recent years; what is more! the boundary between them has become increasingly blurred. This was apparent during the MSRI program! where there were a number of joint seminars on subjects of common interest: birational geometry! D-modules! invariant theory! matrix factorizations! noncommutative resolutions! singularity categories! support varieties! and tilting theory! to name a few. These volumes reflect the lively interaction between the subjects witnessed at MSRI. The Introductory Workshops and Connections for Women Workshops for the two programs included lecture series by experts in the field. The volumes include a number of survey articles based on these lectures! along with expository articles and research papers by participants of the programs. Zusammenfassung This book surveys fundamental current topics in these two areas of research! emphasising the lively interaction between them. Volume 1 contains expository papers ideal for those entering the field. Inhaltsverzeichnis Preface David Eisenbud, Srikanth B. Iyengar, Anurag K. Singh, J. Toby Stafford and Michel Van den Bergh; 1. Growth functions Jason P. Bell; Syzygies, finite length modules, and random curves Christine Berkesch and Frank-Olaf Schreyer; 2. Vector bundles and ideal closure operations Holger Brenner; 3. Hecke algebras and symplectic reflection algebras Maria Chlouveraki; 4. Limits in commutative algebra and algebraic geometry Steven Dale Cutkosky; 5. Introduction to uniformity in commutative algebra Craig Huneke and Claudiu Raicu; 6. Noncommutative motives and their applications Matilde Marcolli and Gonçalo Tabuada; 7. Infinite graded free resolutions Jason McCullough and Irena Peeva; 8. Poincaré-Birkhoff-Witt theorems Anne V. Shepler and Sarah Witherspoon; 9. Frobenius splitting in commutative algebra Karen E. Smith and Wenliang Zhang; 10. From Briançon-Skoda to Scherk-Varchenko Duco van Straten; 11. The interplay of algebra and geometry in the setting of regular algebras Michaela Vancliff; 12. Survey on the D-module f s Uli Walther, with an appendix by Anton Leykin; 13. Introduction to derived categories Amnon Yekutieli....
