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.
 

These talks provide an introduction to mapping class groups of infinite-type surfaces. In the first talk, we survey basic examples of infinite-type surfaces and then turn to general features of their mapping class groups, emphasizing the interplay between the topology of the mapping class group and its algebraic structure.

The second talk focuses on geometric aspects. We first present the general framework of coarsely bounded generation and the resulting large-scale geometry of CB-generated groups, and then illustrate this framework through concrete examples arising from big mapping class groups.

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.

These talks provide an introduction to mapping class groups of infinite-type surfaces. In the first talk, we survey basic examples of infinite-type surfaces and then turn to general features of their mapping class groups, emphasizing the interplay between the topology of the mapping class group and its algebraic structure.

The second talk focuses on geometric aspects. We first present the general framework of coarsely bounded generation and the resulting large-scale geometry of CB-generated groups, and then illustrate this framework through concrete examples arising from big mapping class groups.

This minicourse focuses on counting and equidistribution in homogeneous spaces. We will discuss results for rank-one settings in detail, while briefly surveying developments in the higher-rank case.

The first lecture explains how to derive Haar mixing from Bowen-Margulis-Sullivan mixing by analyzing transversal intersections, a technique developed by Roblin. The second lecture applies these mixing results to study the distribution of Jordan and Cartan spectra, which correspond to counting closed geodesics and orbit points, respectively.

It is tempting to look for a unified flow-theoretic framework for the Thurston norm that generalizes the fibered picture from Talk 1. I will discuss some fundamental results of Mosher in this direction, as well as his Norm and Flow Finiteness Conjecture, which asserts that the Thurston norm of an oriented irreducible atoroidal 3-manifold can be computed by finitely many pseudo-Anosov flows. To attack this conjecture, one needs a way of building pseudo-Anosov flows, which will lead us to the famous Gabai-Mosher construction of pseudo-Anosov flows from sutured manifold hierarchies. I will present a few examples of this, and mention some more recent developments if time allows.

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.