Talks

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Geometry, Groups and Dynamics (GGD) - 2017
Monday, 06 November 2017
Time Speaker Title Resources
09:30 to 10:45 Harish Seshadri (IISc, India) Crash course in Riemanian geometry - Lecture 1

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.
 
10:45 to 11:45 -- Tea
11:15 to 12:30 Mahan Mj (TIFR, India) Introduction to hyperbolic groups - Lecture 1
 
12:30 to 14:30 -- Lunch
14:30 to 15:45 Athanase Papadopoulos (University of Strasbourg, France) Introduction to spherical geometry I
15:45 to 16:15 -- Tea
16:15 to 17:30 Norbert A' Campo (University of Basel, Switzerland) Riemann surfaces: algebra, analysis, geometry - Lecture 1
 
Tuesday, 07 November 2017
Time Speaker Title Resources
09:30 to 10:45 Harish Seshadri (IISc, India) Crash course in Riemanian geometry - Lecture 2
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Mahan Mj (TIFR, India) Introduction to hyperbolic groups - Lecture 2
 
12:30 to 14:30 -- Lunch
14:30 to 15:45 Athanase Papadopoulos (University of Strasbourg, France) Introduction to spherical geometry II
15:45 to 16:15 -- Tea
16:15 to 17:30 Norbert A' Campo (University of Basel, Switzerland) Riemann surfaces: algebra, analysis, geometry - Lecture 2
 
Wednesday, 08 November 2017
Time Speaker Title Resources
09:30 to 10:45 Harish Seshadri (IISc, India) Crash course in Riemanian geometry - Lecture 3
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Mahan Mj (TIFR, India) Introduction to hyperbolic groups - Lecture 3
 
12:30 to 14:30 -- Lunch
14:30 to 15:45 Subhojoy Gupta (IISc, India) Hyperbolic surfaces and their Teichmüller spaces - Lecture 1
 
15:45 to 16:15 -- Tea
Thursday, 09 November 2017
Time Speaker Title Resources
09:30 to 10:45 Pranab Sardar (IISERM, India) Geometry of the symmetric space SL(n,R)/SO(n,R) - Lecture 1

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).
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Subhojoy Gupta (IISc, India) Hyperbolic surfaces and their Teichmüller spaces - Lecture 2

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.
 
12:30 to 14:30 -- Lunch
14:30 to 15:45 Pralay Chatterjee (IMSc, India) Symmetric spaces - Lecture 1

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.
 
15:45 to 16:15 -- Tea
16:15 to 17:30 Ken'ichi Ohshika (Osaka University, Japan) Boundaries of quasi-Fuchsian spaces and continuous/discontinuous phenomena - Lecture 1
 
Friday, 10 November 2017
Time Speaker Title Resources
09:30 to 10:45 Pranab Sardar (IISERM, India) Geometry of the symmetric space SL(n,R)/SO(n,R) - Lecture 2
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Subhojoy Gupta (IISc, India) Hyperbolic surfaces and their Teichmüller spaces - Lecture 3
 
12:30 to 14:30 -- Lunch
14:30 to 15:45 Pralay Chatterjee (IMSc, India) Symmetric spaces - Lecture 2

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.
 
15:45 to 16:15 -- Tea
Saturday, 11 November 2017
Time Speaker Title Resources
09:30 to 10:45 Pranab Sardar (IISERM, India) Geometry of the symmetric space SL(n,R)/SO(n,R) - Lecture 3

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.
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Pralay Chatterjee (IMSc, India) Symmetric spaces - Lecture 3

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.
 
12:30 to 14:30 -- Lunch
Monday, 13 November 2017
Time Speaker Title Resources
09:20 to 09:30 Rajesh Gopakumar (ICTS, India) Welcome remarks
09:30 to 10:45 Misha Kapovich (UC Davis, USA) Discrete Isometry Group of Higher Rank Symmetric Spaces - Lecture 1
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Michah Sageev (Technion – Israel Institute of Technology, Israel) CAT(0) cube complexes and group theory - Lecture 1
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Rostislav Grigorchuk (Texas A&M University, USA) Growth of finitely generated groups and related topics - Lecture 1
 
15:15 to 15:45 -- Tea
15:45 to 17:00 Ken'ichi Ohshika (Osaka University, Japan) Boundaries of quasi-Fuchsian spaces and continuous/discontinuous phenomena - Lecture 2
 
Tuesday, 14 November 2017
Time Speaker Title Resources
09:30 to 10:45 Misha Kapovich (UC Davis, USA) Discrete Isometry Group of Higher Rank Symmetric Spaces - Lecture 2
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Michah Sageev (Technion – Israel Institute of Technology, Israel) CAT(0) cube complexes and group theory - Lecture 2
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Rostislav Grigorchuk (Texas A&M University, USA) Growth of finitely generated groups and related topics - Lecture 2
 
15:15 to 15:45 -- Tea
15:45 to 17:00 Mladen Bestvina (University of Utah, USA) Constructing group actions on quasi-trees and applications - Lecture 1
 
Wednesday, 15 November 2017
Time Speaker Title Resources
09:30 to 10:45 Michah Sageev (Technion – Israel Institute of Technology, Israel) CAT(0) cube complexes and group theory - Lecture 3

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.
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Francois Labourie (Université Côte d'Azur, France) Mini Course 1: Dynamics of Anosov representations - Lecture 1

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.
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Rostislav Grigorchuk (Texas A&M University, USA) Growth of finitely generated groups and related topics - Lecture 3
 
15:15 to 15:45 -- Tea
Thursday, 16 November 2017
Time Speaker Title Resources
09:30 to 10:45 Misha Kapovich (UC Davis, USA) Discrete Isometry Group of Higher Rank Symmetric Spaces - Lecture 3
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Francois Labourie (Université Côte d'Azur, France) Mini Course 1: Dynamics of Anosov representations - Lecture 2

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.
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Jason Behrstock (CUNY, USA) Hierarchy Hyperbolic Spaces - Lecture 1
 
15:15 to 15:45 -- Tea
15:45 to 17:00 Mladen Bestvina (University of Utah, USA) Constructing group actions on quasi-trees and applications - Lecture 2

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.
 
Friday, 17 November 2017
Time Speaker Title Resources
09:30 to 10:45 Misha Kapovich (UC Davis, USA) Discrete Isometry Group of Higher Rank Symmetric Spaces - Lecture 4
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Francois Labourie (Université Côte d'Azur, France) Mini Course 1: Dynamics of Anosov representations - Lecture 3
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Jason Behrstock (CUNY, USA) Hierarchy Hyperbolic Spaces - Lecture 2
 
15:15 to 15:45 -- Tea
Saturday, 18 November 2017
Time Speaker Title Resources
09:30 to 11:00 Anish Ghosh (TIFR, India) Dynamical systems on homogenous spaces and number theory
   
11:00 to 11:30 -- Tea
11:30 to 12:30 Athanase Papadopoulos (University of Strasbourg, France) On some theorems on spherical geometry from Menelaus' Spherics
12:30 to 13:00 -- Discussions
13:00 to 14:00 -- Lunch
Monday, 20 November 2017
Time Speaker Title Resources
09:30 to 10:45 Shrikrishna G Dani (IITB, India) Hyperbolic geometry, the modular group and Diophantine approximation - Lecture 1

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.
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Francois Labourie (Université Côte d'Azur, France) Mini course 2: Introduction to Higgs bundles - Lecture 1

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.
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Athanase Papadopoulos (University of Strasbourg, France) The arc metric on Teichmüller space - Lecture 3

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.
15:15 to 15:45 -- Tea
15:45 to 17:00 Norbert A' Campo (University of Basel, Switzerland) Riemann surfaces: algebra, analysis, geometry - Lecture 3

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,.....
 
Tuesday, 21 November 2017
Time Speaker Title Resources
09:30 to 10:45 Shrikrishna G Dani (IITB, India) Hyperbolic geometry, the modular group and Diophantine approximation - Lecture 2

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.
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Francois Labourie (Université Côte d'Azur, France) Mini course 2: Introduction to Higgs bundles - Lecture 2
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Athanase Papadopoulos (University of Strasbourg, France) The arc metric on Teichmüller space - Lecture 4

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.
15:15 to 15:45 -- Tea
15:45 to 17:00 Norbert A' Campo (University of Basel, Switzerland) Riemann surfaces: algebra, analysis, geometry - Lecture 4

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,.....
 
Wednesday, 22 November 2017
Time Speaker Title Resources
09:30 to 10:45 Keith Burns (Northwestern University, USA) Ergodicity of the Weil-Petersson geodesic flow - Lecture 1
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Francois Labourie (Université Côte d'Azur, France) Mini course 2: Introduction to Higgs bundles - Lecture 3
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Jason Behrstock (CUNY, USA) Hierarchy Hyperbolic Spaces - Lecture 3
 
15:15 to 15:45 -- Tea
Thursday, 23 November 2017
Time Speaker Title Resources
09:30 to 10:45 Keith Burns (Northwestern University, USA) Ergodicity of the Weil-Petersson geodesic flow - Lecture 2
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Pierre Will (Institut Fourier, USA) Discrete groups in complex hyperbolic geometry - Lecture 1
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Athanase Papadopoulos (University of Strasbourg, France) Transitional geometry - Lecture 5

I will give an introduction to transitional geometry, that is, the subject of continuous passages between geometries.
15:15 to 15:45 -- Tea
15:45 to 17:00 Norbert A' Campo (University of Basel, Switzerland) Riemann surfaces: algebra, analysis, geometry - Lecture 5
 
Friday, 24 November 2017
Time Speaker Title Resources
09:30 to 10:45 Keith Burns (Northwestern University, USA) Ergodicity of the Weil-Petersson geodesic flow - Lecture 3
 
10:45 to 11:15 -- Tea
11:15 to 12:30 Pierre Will (Institut Fourier, USA) Discrete groups in complex hyperbolic geometry - Lecture 2

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.
 
12:30 to 14:00 -- Lunch
14:00 to 15:15 Michael Wolf (Rice University, USA) Harmonic Maps between surfaces and Teichmuller theory - Lecture 1
 
15:15 to 15:45 -- Tea
15:45 to 17:00 Michael Wolf (Rice University, USA) Harmonic Maps between surfaces and Teichmuller theory - Lecture 2