## Pierre Dehornoy

**Immersions of a disc in the plane**

P. Dehornoy планирует провести 3 занятия.

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:

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

**Organization Committee e-mail:**

dubna@mccme.ru