Schedule, Notes/Handouts & Videos

The topic of this minicourse is the geometry of the higher rank real semisimple Lie groups, their subgroups and their homogeneous spaces. We will focus on concrete examples like the group of real unimodular matrices. We will  explain the classical structure theorems for these groups and their subgroups.

As an application, we will introduce various natural exponents, like the relative exponential growth or the decorrelation speed, associated to such homogeneous spaces. We will see that these exponents are always equal. This is a joint result with Siwei Liang that solves questions raised by Kobayashi and extends previous results of Patterson, Sullivan, Corlette, Leuzinger, Edwards-Oh and Lutsko-Weich-Wolf.
 

Foliations, flows, and many other dynamical structures on (hyperbolic) 3-manifolds may be tamed and understood via so-called “universal circles”. Conversely, thinking about such structures in terms of universal circles motivates the definition and study of new objects (Zippers, CaTherine wheels, etc) and creates bridges to adjacent fields of mathematics (uniform quasimorphisms, expanding Thurston maps, SLE(8) curves etc). I will define as many of these structures, and explain as much of the network of sometimes subtle and sometimes conjectural relationships between them, as time allows. No previous knowledge of these subjects is assumed.

The topic of this minicourse is the geometry of the higher rank real semisimple Lie groups, their subgroups and their homogeneous spaces. We will focus on concrete examples like the group of real unimodular matrices. We will  explain the classical structure theorems for these groups and their subgroups.

As an application, we will introduce various natural exponents, like the relative exponential growth or the decorrelation speed, associated to such homogeneous spaces. We will see that these exponents are always equal. This is a joint result with Siwei Liang that solves questions raised by Kobayashi and extends previous results of Patterson, Sullivan, Corlette, Leuzinger, Edwards-Oh and Lutsko-Weich-Wolf.

The topic of this minicourse is the geometry of the higher rank real semisimple Lie groups, their subgroups and their homogeneous spaces. We will focus on concrete examples like the group of real unimodular matrices. We will  explain the classical structure theorems for these groups and their subgroups.

As an application, we will introduce various natural exponents, like the relative exponential growth or the decorrelation speed, associated to such homogeneous spaces. We will see that these exponents are always equal. This is a joint result with Siwei Liang that solves questions raised by Kobayashi and extends previous results of Patterson, Sullivan, Corlette, Leuzinger, Edwards-Oh and Lutsko-Weich-Wolf.

Foliations, flows, and many other dynamical structures on (hyperbolic) 3-manifolds may be tamed and understood via so-called “universal circles”. Conversely, thinking about such structures in terms of universal circles motivates the definition and study of new objects (Zippers, CaTherine wheels, etc) and creates bridges to adjacent fields of mathematics (uniform quasimorphisms, expanding Thurston maps, SLE(8) curves etc). I will define as many of these structures, and explain as much of the network of sometimes subtle and sometimes conjectural relationships between them, as time allows. No previous knowledge of these subjects is assumed.