Symmetric Functions 2001: Surveys of Developments and Perspectives : Proceedings of the NATO Advanced Study Instutute on Symmetric Functions 2001: Surveys of Developments and Perspectives Cambridge, U.K. 25 June–6 July 2001

This book surveys recent developments and outlines research prospects in various fields, the fundamental questions of which can be stated in the language of symmetric functions. Interdisciplinary interconnections are emphasized. This volume contains the proceedings of the NATO Advanced Study Institute "Symmetric Functions 2001: Surveys of Developments and Per­ spectives", held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, during the two weeks 25 June - 6 July 2001. The objective of the ASI was to survey recent developments and outline research perspectives in various fields, for which the fundamental questions can be stated in the language of symmetric functions (along the way emphasizing interdisciplinary connections). The instructional goals of the event determined its format: the ASI consisted of about a dozen mini-courses. Seven of them served as a basis for the papers comprising the current volume. The ASI lecturers were: Persi Diaconis, William Fulton, Mark Haiman, Phil Hanlon, Alexander Klyachko, Bernard Leclerc, Ian G. Macdonald, Masatoshi Noumi, Andrei Okounkov, Grigori Olshanski, Eric Opdam, Ana­ toly Vershik, and Andrei Zelevinsky. The organizing committee consisted of Phil Hanlon, Ian Macdonald, Andrei 0 kounkov, G rigori 0 lshanski (co-director), and myself ( co-director). The original ASI co-director Sergei Kerov, who was instrumental in determining the format and scope of the event, selection of speakers, and drafting the initial grant proposal, died in July 2000. Kerov's mathemat­ ical ideas strongly influenced the field, and were presented at length in a number of ASI lectures. A special afternoon session on Monday, July 2, was dedicated to his memory Front Matter....Pages i-xiii Notes on Macdonald Polynomials and the Geometry of Hilbert Schemes....Pages 1-64 The Laplacian Method....Pages 65-91 Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams....Pages 93-151 Symmetric Functions and the Fock Space....Pages 153-177 An Introduction to Birational Weyl Group Actions....Pages 179-222 Symmetric Functions and Random Partitions....Pages 223-252 From Littlewood-Richardson Coefficients to Cluster Algebras in Three Lectures....Pages 253-273 Proceedings of the NATO Advanced Study Institute, held in Cambridge, UK, from 25th June to 6th July, 2001