Panorama of Mathematics - Manjul Bhargava

Explore the Birch and Swinnerton-Dyer Conjecture in this illuminating conference talk by renowned mathematician Manjul Bhargava. Delve into the world of rational solutions to polynomial equations, starting with the fundamental theorem about rational points on conics. Discover the intriguing structure of rational points on cubic curves and learn about the group law on elliptic curves. Gain insights into the modern formulation of the Birch and Swinnerton-Dyer Conjecture, one of the Clay Millennium Problems, and examine the current state of knowledge surrounding this important mathematical challenge.

Intro
The Clay Millennium Problems
Finding rational solutions to polynomial equations
How to find all rational points on the unit circle
Fundamental Theorem about Rational Points on Conics
What about finding rational solutions to two-variable equations of degree three?
The amazing structure of rational points on cubic curves
The group law on eliptic curves
The Birch and Swinnerton-Dyer Conjecture (cont'd)
Modern formulation
What is known about the BSD Conjecture?