Immersions of a disc in the plane. Lecture 1

Imagine an elastic disc floating freely in space. It can take many forms. Now project this disc on a horizontal plane. Which form do you see? Almost anything is possible. The projection of the boundary of this disc is the projection of a circle in the space, and can be any closed curve.
Now suppose that your floating disc is never vertical, and look at his projection on a horizontal plane. Can we obtain as many forms as before? No. For example the projection of the boundary cannot be a figure-eight.
Which curves in the plane can be obtained by projecting the boundary of such a never-vertical-disc? The answer of this question is a theorem of Samuel Blank. Explaining and proving it is the main goal of this course. It is also a pretext to introduce and play with (one of) the most fundamental object of modern geometry and topology: the «fundamental group» of a space.
There are no particular prerequisites for this course, except to understand a bit of english! The program should be roughly:

• Definition of embeddings and immersions, analysis of the problem of embedding a disc in the plane. Introduction to the fundamental group.
• Statement of the theorem of S. Blank, proof of some easy cases.
• End of the proof.
• (if time permits) Introduction to Arnol’d’s invariants of planar curves.