Dynamics, Geometry, Number Theory

Description

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. David Fisher is the Ruth N. Halls Distinguished Professor of Mathematics at Indiana University, Bloomington. Dmitry Kleinbock is professor of mathematics at Brandeis University. Gregory Soifer is professor emeritus of mathematics at Bar-Ilan University, Israel.

Introduction
David FisherPART I || Arithmeticity, superrigidity, normal subgroups1. Superrigidity, arithmeticity, normal subgroups: results, ramifications, and directions 
David Fisher2. An extension of Margulis’s superrigidity theorem
Uri Bader and Alex Furman3. The normal subgroup theorem through measure rigidity
Aaron Brown, Federico Rodriguez Hertz, and Zhiren WangPART II || Discrete subgroups4. Proper actions of discrete subgroups of affine transformations
Jeffrey Danciger, Todd A. Drumm, William M. Goldman, and Ilia Smilga5. Maximal subgroups of countable groups: a survey
Tsachik Gelander, Yair Glasner, and Gregory SoiferPART III || Expanders, representations, spectral theory6. Tempered homogeneous spaces II
Yves Benoist and Toshiyuki Kobayashi7. Expansion in simple groups
Emmanuel Breuillard and Alexander Lubotzky8. Elements of a metric spectral theory
Anders KarlssonPART IV || Homogeneous dynamics9. Quantitative nondivergence and Diophantine approximation on manifolds 
Victor Beresnevich and Dmitry Kleinbock10. Margulis functions and their applications
Alex Eskin and Shahar Mozes11. Recent progress on rigidity properties of higher rank diagonalizable actions and applications
Elon Lindenstrauss12. Effective arguments in unipotent dynamics
Manfred Einsiedler and Amir Mohammadi13. Effective equidistribution of closed hyperbolic subspaces in congruence quotients of hyperbolic spaces
Manfred Einsiedler and Philipp Wirth14. Dynamics for discrete subgroups of SL2(C)
Hee Oh