Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers by Robert J. Zimmer (PDF)

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Robert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program. Zimmer’s ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology. In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program.

User’s Reviews

Editorial Reviews: Review “Zimmer is one of the most influential contemporary American mathematicians. The corpus of Zimmer’s contributions stands out by its coherence and its grand vision. Much more than being a strong problem-solver, more even than being a theory-builder, Zimmer is a mathematician with an overarching sense of the destination, with a domineering command of all relevant available techniques, with the inspiration to bring to life those techniques that were not available yet, with, in conclusion, and to use a trivial formulation, a program.” — Nicolas Monod, École Polytechnique Fédérale de Lausanne“The classic works of Furstenberg, Mostow, and especially Margulis in the ’60s and ’70s led to a satisfactory understanding of the linear actions of lattices in high rank simple Lie groups. In the ’80s, Zimmer conceived a profoundly ambitious plan for understanding non-linear actions, known today as the Zimmer program. The present book provides the blue prints of the plan whose resultant architecture has inspired an entire generation of mathematicians working on Lie groups, differential geometry, ergodic theory, dynamics, arithmetic groups and more. This publication comes out at a perfect time when a number of his disciples have recently succeeded in proving some of his primary conjectures.” — Alexander Lubotzky, Einstein Institute of Mathematics, Hebrew University“This selection of Zimmer’s mathematical papers allows the reader to follow the natural evolution of his far-reaching results and ideas on groups actions in geometry and topology. The final two chapters (written by David Fisher) on the ‘Zimmer Program’ and its recent progress, highlight the broad and profound impact that Zimmer’s works continue to enjoy.” — Peter Sarnak, Institute for Advanced Study About the Author David Fisher is the Ruth N. Halls Distinguished Professor of Mathematics at Indiana University, Bloomington.

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