A quintessential theme in number theory going back to Diophantus is to describe integer solutions of polynomial equations with the use of arithmetic and geometric tools. Integer solutions of an irreducible homogenous polynomial equation $f(x,y,z)\in \mathbb{Z}[x,y,z]$ are rational points on a curve in $P^3$ ( for instance, a celebrated example is Fermat’s equation $f(x,y,z):=x^n+y^n-z^n=0$ is a plane curve). It’s easy to describe the case when $f$ is a conic (ie, degree 2). However, when $f$ is a cubic this is related to the million dollar problem, the Birch and Swinnerton-Dyer Conjecture, which essentially states that the panorama for all smooth curves with affine equation $y^2=x^3-ax-b$ for $a,b\in \mathbb{Q}$ lies in analysis. In this talk we wish to describe the statement of the conjecture and some directions in number theory it has sparked.
Note: No number theoretic back ground is prerequisite for most of this talk.