ICTS

Riemann surfaces: algebra, analysis, geometry

Riemann started the study of the possibility of finding meromorphic functions with prescribed truncated Laurent expansions at its zero's and poles on compact Riemann surfaces. This study was seminal and has forced discoveries in many, perhaps all, branches of mathematics. Here only some key words: topology, manifold, dierential forms, cohomology, hyperbolic geometry, Gauss-Bonnet theorem, harmonic analysis, fundamental group, Teichmueller space, Chow's Theorem, ... The lectures will have as goal the so-called Riemann Existence Theorem, Uniformisation Theorem for compact connected Riemann surfaces of genus g > 1, the Universal Curve,.....
 

Symmetric spaces

Starting with the basics of symmetric spaces and examples, we will explore Chapter 2 of the book "Geometry of nonpositively curved manifolds" by P. Eberlein which deals with the structure of symmetric spaces of non-compact type.
 

Hyperbolic surfaces and their Teichmüller spaces

In the first talk, we shall introduce the Teichmüller space T of a compact oriented surface S as the deformation space of hyperbolic structures on S. In the second talk, we shall discuss the relation with surface group representations to PSL(2,R) and in the case S is closed, identify T with a component comprising discrete and faithful representations. In the final talk, we shall introduce the mapping class group of S and discuss its action on Teichmüller space T.

I will only assume some basic familiarity with hyperbolic plane and its isometries.
 

Introduction to hyperbolic groups

Hyperbolic metric spaces were discovered by Gromov in the 80's to give a unified treatment of manifolds of negative curvature and discrete groups satisfying certain combinatorial conditions. We shall describe various equivalent notions of hyperbolicity. We shall also describe the Gromov boundary of such spaces. (Reference: Bridson-Haefliger--Spaces of non-positive curvature).
 

Boundaries of quasi-Fuchsian spaces and continuous/discontinuous phenomena

In this series talks, I shall first explain what kind of Kleinian groups appear on boundaries of quasi-Fuchsian spaces. Relying on examples given in the first part, I shall next show that as quasi-Fuchsian groups converge to a Kleinian group on the boundary, there are things which vary continuously and other things which vary discontinuously. In the last part I shall illustrate that this difference can be understood using geometric limits.
 

Lecture 1: Introduction to spherical geometry I

Lecture 2: Introduction to spherical geometry II

I will prove some basic theorems of spherical geometry : In a triangle, the angle sum is greater than two right angles ; the segment joining the midpoints of the two legs is greater than half of the basis, and I will give two formulas for the area (one in terms of angles, and one in terms of side lengths). I will discuss the relation with hyperbolic geometry, and more generally with Funk and Hilbert geometries.

Geometry of the symmetric space SL(n,R)/SO(n,R)

I will discuss the geometry of SL(n,R)/SO(n,R) following the book Metric Spaces of Non-positive Curvature by Bridson-Haefliger (Part II, Chapter 10). Starting from the definition of the metric we will prove that it is a CAT(0) space; we will describe the flats, and Weyl chambers etc. Time permitting we will touch on the Tits boundary of this space.
 

Crash course in Riemanian geometry

I will discuss some basic techniques and results in Riemannian geometry relevant to the study of negatively curved manifolds. These will include Jacobi fields, the Cartan-Hadamard theorem and the Cartan-Ambrose-Hicks theorem.

The text " Riemannian Geometry" By Gallot, Hulin and Lafontaine will be the reference.

 

Hierarchy Hyperbolic Spaces

In the first lecture we will introduce the mapping class group (MCG) of a surface and discuss the curve complex of a surface. This leads to several very powerful tools developed by Masur, Minsky, Behrstock, and others which we will survey. In the second lecture we will discuss fundamental groups of compact cube complexes, focussing in particular on the interesting special case of right angled Artin groups (RAAGs). After introducing these groups, we will begin to discuss recent tools developed by Behrstock-Hagen-Sisto which are analogues of the machinery discussed for the mapping class group in the first lecture; we will give details of the tools for RAAGs and describe the general framework of a hierarchically hyperbolic space (HHS). In the third lecture we will outline some of the techniques for working with HHS and give a sketch of the proof of some applications, including the quasi-flats theorems, which proves a number of outstanding conjectures.

References:

1. This is a survey which is a good introduction to the topics which will be covered in this mini-course: "What is a hierarchically hyperbolic space?", by Alessandro Sisto, https://arxiv.org/abs/1707.00053. 
2. This paper won't be discussed explicitly, but it contains and applies a number of the topics discussed in the first lecture, so is a good reference to be aware of. "Geometry and rigidity of mapping class groups", by Jason Behrstock, Bruce Kleiner, Yair Minsky, and Lee Mosher, https://arxiv.org/abs/0801.2006
3. The second lecture will be cover several of the topics of this paper in the special case of RAAGs: Hierarchically hyperbolic spaces I: curve complexes for cubical groups, by Jason Behrstock, Mark F. Hagen, and Alessandro Sisto, https://arxiv.org/abs/1412.2171
4. The main results of this paper will be discussed in lecture 3: "Quasiflats in hierarchically hyperbolic spaces", by Jason Behrstock, Mark Hagen, and Alessandro Sisto, https://arxiv.org/abs/1704.04271
   
    

Constructing group actions on quasi-trees and applications

Starting from relatively simple axioms one can construct a quasi-tree in a natural way. These axioms turn out to be satisfied in a variety of situations that arise in geometric group theory, most notably in the setting of Masur-Minsky subsurface projections, or in the presence of rank 1 group elements. The original construction is presented in arXiv:1006.1939, and it is joint work with Ken Bromberg and Koji Fujiwara. There is a simplification of the construction which I will present in the minicourse, and it is a joint work with Bromberg, Fujiwara, and Alessandro Sisto.
 

Growth of finitely generated groups and related topics

I will explain what is the growth of a group (semigroup, algebra,...), the relation between growth of a group and of Riemannian manifold. Describe the main results about growth of groups from Shwartz and Milnor's contributions (50th-60th of 20th century) till our days. The main focus will be given to groups of intermediate growth (between polynomial and exponential). Starting from my original construction of such groups (that answered the question of Milnor and solved some other problems) I will finish by recent constructions by Bartholdi-Erschler and by Nekrashevych. Also such topics as actions on rooted trees, self-similar groups, automaton groups, amenable groups, and iterated monodromy groups will appear occasionally.
 

Discrete subgroups of higher rank Lie groups

I will talk about recent advances in the theory of discrete subgroups of higher rank Lie groups, which exhibit some "rank one" behavior and related aspects of coarse geometry of higher rank symmetric spaces. This is based mostly on my work with Bernhard Leeb and Joan Porti.
 

Mini Course 1: Dynamics of Anosov representations

In this series of lectures I will give the definition of Anosov subgroups of linear groups emphasizing their dynamical nature. In particular, I will explain that they are associated to currents and that they have natural geodesic flows with dynamical properties. My talk intend to be elementary, starting from Anosov representations of the group Z.

Mini course 2: Introduction to Higgs bundles

In this series of lecture, I will first start with very elementary classical result on line bundles on Riemann surfaces. Following this historical point of view, I will move to the definition of Higgs bundles. My goal is to motivate and explain the definition and state the major theorems of the field.
 

Boundaries of quasi-Fuchsian spaces and continuous/discontinuous phenomena

In this series talks, I shall first explain what kind of Kleinian groups appear on boundaries of quasi-Fuchsian spaces. Relying on examples given in the first part, I shall next show that as quasi-Fuchsian groups converge to a Kleinian group on the boundary, there are things which vary continuously and other things which vary discontinuously. In the last part I shall illustrate that this difference can be understood using geometric limits.
 

CAT(0) cube complexes and group theory

CAT(0) cube complexes are a particular class of CAT(0) spaces that carry a combinatorial structure which gives them the look and feel of “generalized trees". We will discuss the st ructure of CAT(0) cube complexes, what this structure tells us about the groups that act on them, and how to get build actions of groups on such complexes.
 

Dynamical systems on homogenous spaces and number theory

I will survey some of S. G. Dani's many influential contributions to the above subject.
 

On some theorems on spherical geometry from Menelaus' Spherics (November 18 Special talk)

The « Spherics » by Menelaus of Alexandria (1st-2nd c. A.D.) is the most important book ever written on spherical geometry. It is a profound work. It contains 91 propositions, and some of them are very difficult to prove. An edition, from Arabic texts (the Greek original does not survive), is being published now by De Gruyter, in their series Scientia Graeco-Arabica, No. 21.

Link

This publication contains in particular the first English translation of Menelaus' treatise. In this talk, I will explain some of the major theorems on spherical geometry contained in this work. 

Hierarchy Hyperbolic Spaces

In the first lecture we will introduce the mapping class group (MCG) of a surface and discuss the curve complex of a surface. This leads to several very powerful tools developed by Masur, Minsky, Behrstock, and others which we will survey. In the second lecture we will discuss fundamental groups of compact cube complexes, focussing in particular on the interesting special case of right angled Artin groups (RAAGs). After introducing these groups, we will begin to discuss recent tools developed by Behrstock-Hagen-Sisto which are analogues of the machinery discussed for the mapping class group in the first lecture; we will give details of the tools for RAAGs and describe the general framework of a hierarchically hyperbolic space (HHS). In the third lecture we will outline some of the techniques for working with HHS and give a sketch of the proof of some applications, including the quasi-flats theorems, which proves a number of outstanding conjectures.

References:

1. This is a survey which is a good introduction to the topics which will be covered in this mini-course: "What is a hierarchically hyperbolic space?", by Alessandro Sisto, https://arxiv.org/abs/1707.00053.
2. This paper won't be discussed explicitly, but it contains and applies a number of the topics discussed in the first lecture, so is a good reference to be aware of. "Geometry and rigidity of mapping class groups", by Jason Behrstock, Bruce Kleiner, Yair Minsky, and Lee Mosher, https://arxiv.org/abs/0801.2006
3. The second lecture will be cover several of the topics of this paper in the special case of RAAGs: Hierarchically hyperbolic spaces I: curve complexes for cubical groups, by Jason Behrstock, Mark F. Hagen, and Alessandro Sisto, https://arxiv.org/abs/1412.2171
4. The main results of this paper will be discussed in lecture 3: "Quasiflats in hierarchically hyperbolic spaces", by Jason Behrstock, Mark Hagen, and Alessandro Sisto, https://arxiv.org/abs/1704.04271
   
    

Ergodicity of the Weil-Petersson geodesic flow

The Weil-Petersson metric is a Riemannian metric on the Teichmueller space of a Riemann surface. It is invariant under the action of the mapping class group and descends to a Riemannian metric with volume on the moduli space.

The Weil-Peterson metric has negative sectional curvatures but is incomplete. The curvature and its derivatives blow up as one approaches the boundary of Teichmueller space. The effect of negative curvature on the behaviour of the geodesic flow is well understood. In particular Hopf and Anosov showed that the geodesic flow of a compact manifold with negative curvatures is ergodic. Their results extend to the Weil-Petersson geodesic flow, but the incompleteness of the metric creates considerable additional difficulties.

The lectures will attempt to explain the arguments of Hopf and Anosov and to indicate the additional ideas needed to apply them to the Weil-Petersson geodesic flow.
 

Hyperbolic geometry, the modular group and Diophantine approximation

Study of dynamics of the geodesic flow associated with the modular surface, consisting of the Poincare plane viewed modulo the action of the modular group SL(2, Z) acting as isometries, has been applied to study the distribution of values of quadratic forms at points on the Euclidean plane with integer coordinates. In these talks we shall discuss the framework and some results in this respect. Relation with some other questions in the area of Diophantine approximation will also be discussed.
 

Dynamical systems on homogenous spaces and number theory

I will survey some of S. G. Dani's many influential contributions to the above subject.
 

Mini course 2: Introduction to Higgs bundles

In this series of lecture, I will first start with very elementary classical result on line bundles on Riemann surfaces. Following this historical point of view, I will move to the definition of Higgs bundles. My goal is to motivate and explain the definition and state the major theorems of the field.
 

Lecture 3 and 4 : The arc metric on Teichmüller space

I will give an overview of the arc metric on the Teichmüller space of surfaces with boundary, including a study of its geodesics, its horofunction boundary, and the fact that the arc metric converges in an appropriate way to the Thurston metric of surfaces without boundary.

Lecture 5 : Transitional geometry

I will give an introduction to transitional geometry, that is, the subject of continuous passages between geometries.

Discrete groups in complex hyperbolic geometry

I will discuss the geometry of complex hyperbolic space, and describe examples of dicrete subgroups of its isometry group. I will mainly focus on the complex hyperbolics analogues of quasi Fuchsian groups. My goal is to end up describing recent progresses concerning spherical CR structures on 3-manifolds, which appear on the boundary at infinity of quotients of the complex hyperbolic plane by discrete subgroups.
 

Harmonic Maps between surfaces and Teichmuller theory, I and II

We describe some of the basics of classical Teichmuller theory from the point of view of harmonic maps. The goal is to give an introduction to the Hitchin component of the PSL(2,R) character variety that one would undertake after one has identified a convenient setting and phrasing for the Hitchin equations, which in this case take the form of the 'Bochner equation' for harmonic maps of surfaces. We describe elements of the local deformation theory as well as the asymptotics. We will not assume any prior knowledge of harmonic maps.