An exotic sphere is a manifold which is homeomorphic but not
diffeomorphic to a sphere. These objects caused great surprise among
mathematicians, when they were first exhibited by John Milnor in 1956.
Milnor later went on to win the fields medal for his work on this topic.
In this talk, I will carefully explain what is meant by an exotic
sphere (as well as its significance) and I will provide a step by step
construction of such a sphere in dimension 7. Time permitting, I will
say a few words about the classification problem.