CIMPA SCHOOL 2021 : GROUPS AND GEOMETRY IN KOLKATA
New dates : January 18-29, 2022





Initially planned during the period 18/01/2021-29/01/2021, the pandemic situation led us to postpone the event to 18/01/2022-29/01/2022.



Presentation
Scientific program
Schedule
 Links
 Sponsors





Presentation

Geometric group theory is a relatively new line of research on its own, inspired by pioneering works of M. Dehn, G.D. Mostow and M. Gromov. It is mainly devoted to the study of countable groups by exploring connections between algebraic properties of such groups and geometric properties of spaces on which these groups act, such as the deck transformation group of a Riemannian manifold. Nowadays, geometric group theory is a very active and competitive area of research and is the subject of many conferences and special programs in recent years. Geometric group theory is a very broad area, and this program aims at introducing young students to different aspects of the theory.

Starting from classical results such as the Mostow and Kleiner-Leeb rigidity theorems, we intend to build a body of knowledge to the students by introducing them to some basic material concerning hyperbolic geometry, quasi-isometry invariants and amenability and to several modern aspects of geometric group theory: boundary theory, CAT(0) cube complexes, ergodic theory and representations.

It will therefore teach, to graduate students across mathematical fields, the important trends in geometric group theory so that they can use geometric group theory methods in their respective fields.

Administrative and scientific coordinators

Scientific committee

Genevieve Walsh (Tufts University, USA)
Christophe Pittet (Université d'Aix-Marseille, France)
Tullia Dymarz (University of Wisconsin, USA)
Manjusha Majumdar (University of Calcutta, India)

Contact

Mail to organizers :



Scientific program

Titles of the courses

Course 1: "Prerequisites on Riemannian Geometry", Arindam Bhattacharyya (Jadavpur University, India)
Course 2: "Hyperbolic Geometry and Mostow Rigidity", Kingshook Biswas (Indian Statistical Institute, India) and Krishnendu Gongopadhyay (Indian Institutes of Science Education and Research, India)
Course 4: "Word Hyperbolic Groups", Indira Chatterji (Université Côte d'Azur, France)
Course 5: "Boundaries of Hyperbolic Groups", Genevieve Walsh (Tufts University, USA)
Course 6: "Ergodic Theorems and Unitary Representations of Lie Groups and their Lattices", Christophe Pittet (Université d'Aix-Marseille, France)
Course 7: "Large scale geometry of nilpotent and solvable Lie groups", Tullia Dymarz (University of Wisconsin, USA)
The courses are divided into two categories: four introductory courses and three advanced courses. Training sessions will complement the advanced courses.

Introductory courses

Course 1: "Prerequisites on Riemannian Geometry", Arindam Bhattacharyya (Jadavpur University, India)
Abstract of the course. We shall introduce the basic geometric objects of Riemannian geometry to provide some background: definition of a Riemannian manifold, its associated distance and group of isometries with their elementary properties. Emphasis will be given on examples, including surfaces and Lie groups.

Course 2: "Hyperbolic Geometry and Mostow Rigidity", Kingshook Biswas (Indian Statistical Institute, India) and Krishnendu Gongopadhyay (Indian Institutes of Science Education and Research, India)
Abstract of the course. We shall give a gentle introduction to hyperbolic geometry, insisting on the hyperbolic plane, before considering higher dimensional spaces. The rest of the lectures will be devoted to sketching Sullivan's proof of the Mostow rigidity theorem, which states that if X, Y are closed hyperbolic n-manifolds with n at least three, then any isomorphism between their fundamental groups is induced by an isometry between the manifolds.

Course 3: "Quasi-Isometry Invariants", Peter Haïssinsky (Université d'Aix-Marseille, France)
Abstract of the course. We will discuss the notion of quasi-isometry and present some standard examples of invariants, such as ends, volume growth, finite presentability, etc.. We should also describe Bass-Serre theory from a topological view if time allows.

Course 4: "Word Hyperbolic Groups", Indira Chatterji (Université Côte d'Azur, France)
Abstract of the course. I will discuss Cayley graphs, then growth and hyperbolicity in metric spaces and groups. I will discuss concrete examples to see which ones are hyperbolic and which ones are not.

Advanced courses

Course 5: "Boundaries of Hyperbolic Groups", Genevieve Walsh (Tufts University, USA)
Abstract of the course. We will define and discuss the Gromov boundary of a hyperbolic group. This is an extremely useful topological space associated to a hyperbolic group. We will give many examples of boundaries, and describe some properties of the boundary which are reflected in the group.

Course 6: "Ergodic Theorems and Unitary Representations of Lie Groups and their Lattices", Christophe Pittet (Université d'Aix-Marseille, France)
Abstract of the course. Ergodic theorems can be deduced from operator norm estimates. In the case of Lie groups and their lattices, the theory of unitary representations, in particular estimates of the Harish-Chandra (spherical) functions, lead to very precise convergence rate estimates. (Priority will be given to examples rather than to the general theory.)

Course 7: "Large scale geometry of nilpotent and solvable Lie groups", Tullia Dymarz (University of Wisconsin, USA)
Abstract of the course. The goal of these talks is to introduce students to the basics of large scale geometry and metric structures of nilpotent and solvable Lie groups. The first lecture will review the notion of quasi-isometry and gives basics of nilpotent and solvable groups. The second will focus on aspects of the metric geometry of nilpotent Lie groups. The third will look at the metric geometry of certain family of solvable Lie groups. We will assume some familiarity with Riemannian geometry and the basic definitions of Lie groups and Lie algebras.

Training sessions

Besides the first hour providing some very elementary background on Riemannian geometry, we wish to organize the lectures based on the "learning-by-doing" method. To implement this, the lectures have been grouped in half-days. A portion of the time is used to introduce definitions and some results (with or without proofs). For the rest of the time, the students will be organized into small groups, and we should make them work on examples and find proofs of some results. To have a lot of time is supposed to help the students get acquainted with the objects and methods on their own.



Schedule



DAY-1
18-01-2022
(Tuesday)		 Inauguration
01:30 pm - 2:00 pm 		Arindam Bhattacharyya
02:00 pm 03:00 pm 		Peter Haïssinsky
03:30 pm 04:30 pm 		Krishnendu Gongopadhyay
05:00 pm 06:00 pm
DAY-2
19-01-2022
(Wednesday)		 Peter Haïssinsky & Tutorial
10:30 am 11:30 am 		Krishnendu Gongopadhyay & Tutorial
11:45 am 12:45 pm 		LUNCH
Indira Chatterji & Tutorial
01:30 pm 03:15 pm 		Peter Haïssinsky & Tutorial
03:45 pm 05:30 pm
DAY-3
20-01-2022
(Thursday)		 Indira Chatterji & Tutorial
01:30 pm 03:15 pm 		Peter Haïssinsky & Tutorial
03:45 pm 05:30 pm 		Krishnendu Gongopadhyay & Tutorial
06:00 pm 07:00 pm

DAY-4
21-01-2022
(Friday)		 Indira Chatterji & Tutorial
01:30 pm 03:15 pm 		Krishnendu Gongopadhyay & Tutorial
03:45 pm 06:00 pm


DAY-5
22-01-2022
(Saturday)		 Krishnendu Gongopadhyay & Tutorial
11:30 am 12:30 pm 		LUNCH
Indira Chatterji & Tutorial
01:30 pm 03:15 pm 		Peter Haïssinsky & Tutorial
03:45 pm 04:45 pm 		Krishnendu Gongopadhyay & Tutorial
05:00 pm 06:00 pm
DAY-6
24-01-2022
(Monday)		 Tullia Dymarz & Tutorial
09:00 am 11:00 am 		LUNCH
Christophe Pittet & Tutorial
01:30 pm 03:15 pm 		Kingshook Biswas
03:45 pm 06:00 pm 		Genevieve Walsh
06:30 pm - 07:00 pm
DAY-7
25-01-2022
(Tuesday)		 Tullia Dymarz & Tutorial
09:00 am 11:00 am 		LUNCH
Genevieve Walsh & Tutorial
01:30 pm - 02:30 pm 		Christophe Pittet & Tutorial
03:30 pm 05:15 pm 		Genevieve Walsh
06:30 pm - 07:00 pm
DAY-8
26-01-2022
(Wednesday)		 Tullia Dymarz & Tutorial
09:00 am 11:00 am 		LUNCH
Genevieve Walsh & Tutorial
01:30 pm - 02:30 pm 		Christophe Pittet & Tutorial
03:30 pm 05:15 pm 		Genevieve Walsh
06:30 pm - 07:00 pm
DAY-9
27-01-2022
(Thursday)		 Kingshook Biswas
09:00 pm 11:15 pm 		LUNCH
Genevieve Walsh & Tutorial
01:30 pm - 03:45 pm 		Christophe Pittet & Tutorial
04:00 pm 05:45 pm 		Valedictory Session








Links

  1. CIMPA.




Sponsors

 Cimpa Cimpa Centre international de mathématiques pures et appliquées.

IMU IMU International Mathematical Union

Cimpa Calcutta Mathematical Society

I2M I2M Institut de mathématiques de Marseille