This will be an elementrary introduction to some of the main ideas in symplectic geometry. In this geometry one can make two dimensional measurements, instead of the usual one dimensional measurements such as the length of a line, and its properties are intimately tied to the complex rather than the real numbers. It has a very different flavor from Euclidean geometry, being in many ways rather flabby but also exhibiting robust structures. I will talk about Gromov's nonsqueezing theorem, the symplectic camel, and recent work I have been doing on embedding ellipsoids.
The talk should be accessible to most students.