If you are interested solutions to Diophantine equations then you certainly care about the existence of rational points on nice varieties defined over $\mathbb{Q}$. This proves to be difficult because the field of rational numbers $\mathbb{Q}$ is arithmetically complex. On the other hand, if a variety $X$ over $\mathbb{Q}$ has a rational point then it has a $p$-adic point ($\mathbb{Q}_p$-point) for every prime $p$ (since $\mathbb{Q}\hookrightarrow \mathbb{Q}_p$) and also, of course, a real point. In other words, if $X(\mathbb{Q})\neq \Phi$ then $X(\mathbb{Q}_p)\neq \Phi$ for all primes $p$ and $X(\mathbb{R})\neq \Phi$.

Checking for points in a complete field like $\mathbb{Q}_p$ is a local problem and unlike its global analog, is tractable. If $X$ is nice, it has $\mathbb{Q}_p$-points for all but an effective finite set of primes $p$, and so it's possible to determine if it always has points locally.

It is thereby natural to ask whether having points locally is sufficient for $X$ to have a rational point; the answer is no and many counterexamples exist.

In this talk I will describe a finer condition for the existence of a global, i.e rational point on $X$ by introducing the Brauer-Manin obstruction.

Some background in the language of algebraic geometry is required, this talk will be very expository.