Hyperbolic Geometry Course | IIT Kanpur NPTEL | Prof. Abhijit Pal
Course Details
| Exam Registration | 16 |
|---|---|
| Course Status | Ongoing |
| Course Type | Elective |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 19 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
An Introduction to Hyperbolic Geometry: A 12-Week Postgraduate Journey
Welcome to a detailed exploration of Hyperbolic Geometry, a fascinating branch of non-Euclidean mathematics with profound implications for group theory and manifold study. This 12-week postgraduate course, offered by Prof. Abhijit Pal of IIT Kanpur through NPTEL, is designed to take you from foundational concepts to advanced applications in geometric group theory.
About the Course and Instructor
Prof. Abhijit Pal is a distinguished faculty member in the Department of Mathematics & Statistics at IIT Kanpur. He earned his PhD from ISI Kolkata in 2011 under the guidance of Prof. Mahan Mj. His research expertise lies at the intersection of Hyperbolic Geometry and Geometric Group Theory, with a special focus on hyperbolic and relatively hyperbolic groups. His deep knowledge ensures this course is both rigorous and insightful.
The course delves into hyperbolic geometry, a central object for studying group theory from a geometric viewpoint. You will learn how many surfaces and three-manifolds exhibit hyperbolic structures, making this field crucial for advanced mathematical research.
Who Should Enroll?
Intended Audience: This course is primarily aimed at Postgraduate (PG) and PhD students. However, motivated undergraduate students with a solid background in topology, algebra (group theory), and complex analysis are also welcome to participate.
Prerequisites
To succeed in this course, you should be familiar with:
- Topology
- Algebra (Group Theory)
- Complex Analysis
Need a refresher? Prof. Pal recommends these NPTEL courses:
- Essentials Of Topology: https://nptel.ac.in/courses/111105169
- Algebra - I: https://nptel.ac.in/courses/111106137
- Complex Analysis: https://nptel.ac.in/courses/111106141
Course Layout: A 12-Week Roadmap
The course is systematically divided into three major phases, each building upon the last.
Weeks 1-5: Foundations of Hyperbolic Geometry
We begin by establishing the core models and properties of hyperbolic space.
- Models of Hyperbolic Space: The Upper Half Plane and the Unit Disc with the Poincaré metric.
- Hyperbolic Inner Product and Geodesics: Understanding distance and shortest paths.
- Isometry Groups: Studying the transformations that preserve hyperbolic distance.
- Classification of Isometries: Categorizing these transformations.
- Area of Triangles & Trigonometric Identities: Exploring the surprising properties of hyperbolic triangles, where angles sum to less than 180 degrees.
Weeks 6-8: Groups Acting on Hyperbolic Space
This module connects geometry with algebra through group actions.
- Properly Discontinuous & Cocompact Actions: Key conditions for nice quotient spaces.
- Fuchsian Groups: Discrete groups of isometries of the hyperbolic plane.
- Covering Spaces & Fundamental Regions: Tools for constructing hyperbolic surfaces.
- Tessellation and Poincaré’s Theorem (Statement): How to build a hyperbolic surface from a polygon.
- Closed Curves and Geodesics on Hyperbolic Surfaces: Analyzing paths on these constructed surfaces.
Weeks 9-12: Hyperbolic Groups and Metric Spaces
The final phase generalizes the concepts to abstract metric spaces and groups.
- Slim and Thin Triangles: Defining features of hyperbolic metric spaces.
- Hyperbolic Metric Space: The abstract, large-scale definition of hyperbolicity.
- Exponential Divergence & Stability of Quasi-geodesics: Key geometric properties.
- Hyperbolic Groups: Introducing groups whose Cayley graphs are hyperbolic metric spaces, and exploring their algebraic properties.
Recommended Textbooks & Resources
To complement the lectures, Prof. Pal suggests the following authoritative texts:
| Book Title | Author(s) | Notes |
|---|---|---|
| Fuchsian Groups | Svetlana Katok | The University of Chicago Press. Excellent for the middle part of the course. |
| Metric Spaces of Non-Positive Curvature | M. R. Bridson and A. Haefliger | Springer. A comprehensive reference for the broader theory. |
| A course on geometric group theory | B. H. Bowditch | Freely available online. A great resource for the final section on hyperbolic groups. Link |
Why Study Hyperbolic Geometry?
Hyperbolic geometry is not just a mathematical curiosity. It is a vital language in modern mathematics, essential for understanding:
- The structure of low-dimensional manifolds (per Thurston's Geometrization).
- The large-scale geometry of infinite groups (Geometric Group Theory).
- Complex dynamics and Riemann surface theory.
This course provides the essential toolkit to engage with these active research areas. Under the expert guidance of Prof. Abhijit Pal, you will gain a deep and practical understanding of hyperbolic geometry, from its classical models to its frontier applications in group theory.
Enroll Now →