Symmetry may be very familiar and intuitive to us, but how can we talk about symmetry mathematically? We know that triangles and spheres both possess kinds of symmetry, but how are they related? The mathematics of symmetry can be described by "Group Theory".
To anyone who doesn't use abstract mathematics on a daily basis, the concept of an imaginary number sounds absurd. We can't use them to count things, as we can with natural numbers.
Infinity is still an extremely mysterious concept that we're unable to properly grasp, yet mathematicians have penetrated deeper into the world of infinity than ever before, and there is a wealth of new discoveries. The interesting question is whether these infinities have any relevance to the real physical world.
One of the most puzzling mysteries of modern day Physics is why mathematics is such a useful tool for uncovering the workings of the universe. Although mathematics has always been the only acceptable language in which to phrase serious physics, letting mathematics guide the direction of physics is a late 20th/21st century revolution in scientific thinking.