This class is about imagination and connections of different mathematical concepts: from geometry to group theory to algebra. It will introduce you to the world of symmetries, especially continuous symmetries and its applications.

A few symmetry groups (of small dimensions) -- especially their mutual relations -- will be studied very carefully. These will include rotations, pseudorotations SO(n,1), the Moebius group, SL(n,Z), SL(2,C), Apollonian group, and more.

One of our "toy models" will be the Apollonian gasket -- a certain fractal-like arrangement of circles (or higher-dimensional spheres), which recently enjoys much attention due to its reach connections with different branches of mathematics from geometry to group theory to number theory. Many of the connections were discovered very recently. We will use it as a vehicle to wander through these topics and concepts: Lie groups, Lie algebras, spin structures, inversive geometry, mathematics of relativity, fractals, number theory, etc.

The topics to be covered are of universal importance and go beyond the specific applications.