We recall the definition of the fundamental group develop in the previous lecture then prove that it is indeed a group. Finally, we show that the fundamental group of the circle is isomorphic to Z, the integers.

We give a quick review of group theory then discuss homotopy of paths building up to the definition of the fundamental group.

A nice walk from homotopy to Lie theory to category theory to combinatorics.

Zvezdelina Stankova discusses the raffle function - and her epic proof ends with an interesting connection.

Beautiful, beautiful problem. Abstract algebra, calculus, number theory, and combinatorics all wrapped up into a bow. :)

Is there one ring to rule them all? Kevin Tucker goes deep into the world of mathematical rings, among other things.

Beautiful weaving of geometry, group theory, and visual art.

Nice review of stuff from 18.C06/18.063.