We are pleased to announce the weekly Research Scholar Seminars for this semester.

Seminar Details:

• Speaker: Abhiram H. (BITS Pilani, Pilani Campus)

• Title: Hyperbolic Geometry

• Date & Time: Wednesday, 17 September 2025 , 5:00 – 6:00 PM

• Venue: TBA

Abstract:

The history of geometry is deeply intertwined with the quest to understand the nature of space itself. In ancient Greece, Euclid laid the foundation for geometry by proposing five simple postulates. Among these, the fifth postulate (often referred to as the parallel postulate) presented a unique challenge. It seemed less intuitive than the other axioms, leading mathematicians for centuries to attempt to prove it from the first four. However, these efforts were ultimately in vain. The inability to prove the parallel postulate led to the development of non-Euclidean geometries in the 19th century, most notably hyperbolic geometry. This new
geometry has since found applications in various fields, including physics, where it is now used to describe the structure of space-time itself.

In this talk, I will explore the geometric structure of the upper half-plane, the set of complex numbers with a positive imaginary part. By examining its topology and investigating whether a metric can be naturally induced, imposing some symmetries will reveal how the upper half-plane can serve as a model for hyperbolic geometry. Further, we will analyze the geodesics of the upper half-plane and ultimately address the question: Does the parallel postulate hold in this framework?

All are cordially invited to attend the seminar.