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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.