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Dynamics, Geometry, Number Theory

The Impact of Margulis on Modern Mathematics

This definitive synthesis of mathematician Gregory Margulis’s research brings together leading experts to cover the breadth and diversity of disciplines Margulis’s work touches upon.
This edited collection highlights the foundations and evolution of research by widely influential Fields Medalist Gregory Margulis. Margulis is unusual in the degree to which his solutions to particular problems have opened new vistas of mathematics; his ideas were central, for example, to developments that led to the recent Fields Medals of Elon Lindenstrauss and Maryam Mirzhakhani. Dynamics, Geometry, Number Theory introduces these areas, their development, their use in current research, and the connections between them. Divided into four broad sections—“Arithmeticity, Superrigidity, Normal Subgroups”; “Discrete Subgroups”; “Expanders, Representations, Spectral Theory”; and “Homogeneous Dynamics”—the chapters have all been written by the foremost experts on each topic with a view to making them accessible both to graduate students and to experts in other parts of mathematics. This was no simple feat: Margulis’s work stands out in part because of its depth, but also because it brings together ideas from different areas of mathematics. Few can be experts in all of these fields, and this diversity of ideas can make it challenging to enter Margulis’s area of research. Dynamics, Geometry, Number Theory provides one remedy to that challenge.

496 pages | 22 halftones, 2 tables | 6 x 9

Mathematics and Statistics


"Margulis is without a doubt one of the most influential mathematicians of the past fifty years. The book Dynamics, Geometry, Number Theory is vast in scope and provides an excellent introduction to Margulis's work and the research that it has inspired. It will be of great interest not only to specialists, but to graduate students and researchers interested in ergodic theory, Lie theory, geometry, and number theory."

MAA Reviews

“Margulis is one of the great mathematicians of the twentieth century and the first decades of this century, whose work is central today. This valuable book collects reflections on his work by some of the most prominent scholars in the area. Uniquely broad in scope, the whole collection is very strong, and the whole is greater than the parts. Terrific!”

Shmuel Weinberger, University of Chicago

“A superb contribution in every regard: purely scientifically; expounding upon the many deep works of Margulis and thereby appropriately honoring him and his work; and putting the contributions in context and sorting them in appropriate categories, while explaining the deep connections between them. The intellectual level of this book is astounding. I will recommend it to all my associates, graduate students, postdocs, and other researchers, and surely to my library.”

Ralf Spatzier, University of Michigan

“Margulis’s work has had a tremendous impact on mathematics, and this book will be read by scholars from a broad cross-section of mathematical backgrounds connected to the four sections of the volume and beyond. It will serve as a go-to collection for specialists and graduate students alike.”

Alan Reid, Rice University

Table of Contents

David Fisher

PART I || Arithmeticity, superrigidity, normal subgroups

1. Superrigidity, arithmeticity, normal subgroups: results, ramifications, and directions 
David Fisher
2. An extension of Margulis’s superrigidity theorem
Uri Bader and Alex Furman
3. The normal subgroup theorem through measure rigidity
Aaron Brown, Federico Rodriguez Hertz, and Zhiren Wang

PART II || Discrete subgroups

4. Proper actions of discrete subgroups of affine transformations
Jeffrey Danciger, Todd A. Drumm, William M. Goldman, and Ilia Smilga
5. Maximal subgroups of countable groups: a survey
Tsachik Gelander, Yair Glasner, and Gregory Soifer

PART III || Expanders, representations, spectral theory

6. Tempered homogeneous spaces II
Yves Benoist and Toshiyuki Kobayashi
7. Expansion in simple groups
Emmanuel Breuillard and Alexander Lubotzky
8. Elements of a metric spectral theory
Anders Karlsson

PART IV || Homogeneous dynamics

9. Quantitative nondivergence and Diophantine approximation on manifolds 
Victor Beresnevich and Dmitry Kleinbock
10. Margulis functions and their applications
Alex Eskin and Shahar Mozes
11. Recent progress on rigidity properties of higher rank diagonalizable actions and applications
Elon Lindenstrauss
12. Effective arguments in unipotent dynamics
Manfred Einsiedler and Amir Mohammadi
13. Effective equidistribution of closed hyperbolic subspaces in congruence quotients of hyperbolic spaces
Manfred Einsiedler and Philipp Wirth
14. Dynamics for discrete subgroups of SL2(C)
Hee Oh

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