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.