Schedule, Notes/Handouts & Videos

Ergodic theory studies dynamical systems and group actions in the presence of measures. Typical statements in this field describe existence (or lack) of measurable maps with certain properties, distribution of orbits of typical points etc. It is surprising and fascinating that such phenomena turn out to be very useful in proving some rigidity results in geometry.

In this mini-course I will mention a few recent results related to arithmeticity and linearity, and then focus on a rigidity problem involving delta-hyperbolic spaces.  

Most of the material is based on joint works with Uri Bader.

Thurston’s Geometrization Conjecture, proven in 2003 by Perelman, states that every closed 3-manifold can be decomposed into pieces, each of which admits one of 8 model geometries. We will survey this theorem and its adjacent motivations and corollaries.
 

Ergodic theory studies dynamical systems and group actions in the presence of measures. Typical statements in this field describe existence (or lack) of measurable maps with certain properties, distribution of orbits of typical points etc. It is surprising and fascinating that such phenomena turn out to be very useful in proving some rigidity results in geometry.

In this mini-course I will mention a few recent results related to arithmeticity and linearity, and then focus on a rigidity problem involving delta-hyperbolic spaces.  

Most of the material is based on joint works with Uri Bader.

It is a consequence of the Margulis dichotomy that when an arithmetic hyperbolic manifold contains one totally geodesic hypersurface, it contains infinitely many. Both Reid and McMullen have asked conversely whether the existence of infinitely many geodesic hypersurfaces implies arithmeticity of the corresponding hyperbolic manifold. In this talk, I will discuss recent results answering this question in the affirmative. In particular, I will describe how this follows from a general superrigidity style theorem for certain natural representations of fundamental groups of hyperbolic manifolds. If time permits, I will also discuss a recent extension of these techniques into the complex hyperbolic setting, which requires the aforementioned superrigidity theorems as well as some theorems in incidence geometry due to Pozzetti. This is joint work with Bader, Fisher, and Stover.
 

In 1992, Reid posed the question of whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to Reid's question, Futer and Millichap have recently constructed innitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same rst n geodesic lengths. In the present lecture, we show that this phenomenon is surprisingly common in the arithmetic setting. This talk is based on joint work with B. Linowitz, D. B. McReynolds, and P. Pollack.

The distribution of sums of real-valued random variables is determined by the classical theorems of probability (law of large numbers, central limit theorem). 

Starting in the 1960’s Furstenberg, Oseledets and others have generalized such results tor noncommutative groups, e.g. groups of matrices. 

In this course, we will consider random walks on groups of isometries of delta-hyperbolic spaces,  and establish their asymptotic properties: for instance, sample paths almost surely converge to the boundary and have positive drift. 

In recent years, this has had many applications to low-dimensional topology,  as e.g. the mapping class group and Out(F_n) act on certain (non-locally compact) hyperbolic spaces. We will discuss some such applications and their relations to Teichmuller theory. 

Lecture plan: 

1) Introduction to delta-hyperbolic spaces and random walks

2) The horofunction boundary 

3) Convergence to the boundary and positive drift

4) Genericity of hyperbolic elements

The course is mostly based on my joint work with J. Maher. 

In this lecture, we sketch a proof that there are innitely many k-tuples of arithmetic, hyperbolic 3-orbifolds which are pairwise non-commensurable, have certain prescribed geodesic lengths, and have volumes lying in an interval of bounded length. One of the key ideas stems from the breakthrough work of Maynard and Tao on bounded gaps between primes. We will introduce the Maynard-Tao approach and then discuss how it can be applied in a geometric setting. This talk is based on joint work with B. Linowitz, D. B. McReynolds, and P. Pollack.

Given a variety over a global field K, one can ask how many K-rational points of bounded height it has. However, a lot of interesting spaces are not varieties. For instance, one might want to formulate the same question for stacks, but there are roadblocks to even defining heights on them. This talk explores heights on stacks in the context of counting rational points on moduli spaces of elliptic curves, which can be described classically as quotients of the upper half plane by congruence subgroups. I will talk about height functions on varieties and how they generalize (or don't generalize) to stacks. I will also explain how one can use height machinery developed recently by Ellenberg, Satriano and Zureick-Brown to answer, for certain integers N, the classical question: how many elliptic curves over Q have a rational N isogeny? This is based on joint work with Brandon Boggess. The talk assumes no prior knowledge of stacks or of the arithmetic of elliptic curves. 

Anosov representations are discrete, faithful (or finite-kernel) representations of word hyperbolic groups into semisimple Lie groups, with strong dynamical properties. They were introduced by Labourie in 2006 for fundamental groups of closed negatively-curved manifolds, and generalized by Guichard and Wienhard in 2012. They have been much studied in the past few years, and play an important role in higher Teichmüller-Thurston theory and in recent developments in the theory of discrete subgroups of Lie groups. We will introduce these representations, give examples, and discuss some characterization

I will give an introduction to Property (T) and aTmenability, and illustrate those notions using CAT(0) cubical complexes, that I will define as well.

Anosov representations are discrete, faithful (or finite-kernel) representations of word hyperbolic groups into semisimple Lie groups, with strong dynamical properties. They were introduced by Labourie in 2006 for fundamental groups of closed negatively-curved manifolds, and generalized by Guichard and Wienhard in 2012. They have been much studied in the past few years, and play an important role in higher Teichmüller-Thurston theory and in recent developments in the theory of discrete subgroups of Lie groups. We will introduce these representations, give examples, and discuss some characterization

In real convex projective geometry, simplices have many characteristics of flat geometry. In particular, we will discuss compact convex projective manifolds (in any dimension greater than 2) which have properly embedded codimension-1 simplices, and demonstrate that examining these submanifolds yields strong topological data about the total manifold. This work generalizes a 2006 theorem of Benoist for compact 3-dimensional convex projective manifolds which gives a JSJ decomposition along properly embedded triangles.