Memoirs of the American Mathematical Society 2000; 106 pp; softcover Volume: 146 ISBN10: 0821826573 ISBN13: 9780821826577 List Price: US$49 Individual Members: US$29.40 Institutional Members: US$39.20 Order Code: MEMO/146/695
 We prove a polynomial multiple recurrence theorem for finitely many commuting measure preserving transformations of a probability space, extending a polynomial Szemerédi theorem appearing in [BL1]. The linear case is a consequence of an ergodic IPSzemerédi theorem of Furstenberg and Katznelson ([FK2]). Several applications to the fine structure of recurrence in ergodic theory are given, some of which involve weakly mixing systems, for which we also prove a multiparameter weakly mixing polynomial ergodic theorem. The techniques and apparatus employed include a polynomialization of an IP structure theory developed in [FK2], an extension of Hindman's theorem due to Milliken and Taylor ([M], [T]), a polynomial version of the HalesJewett coloring theorem ([BL2]), and a theorem concerning limits of polynomially generated IPsystems of unitary operators ([BFM]). Readership Researchers interested in measurepreserving transformations, partitions of integers, Ramsey theory, sequences and sets. Table of Contents  Introduction
 Formulation of main theorem
 Preliminaries
 Primitive extensions
 Relative polynomial mixing
 Completion of the proof
 Measuretheoretic applications
 Combinatorial applications
 For future investigation
 Appendix: Multiparameter weakly mixing PET
 References
 Index of notation
 Index
