Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Friday, April 22, 2005, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Uli Wagner (Einstein Inst. of Mathematics, The Hebrew Univ. of Jerusalem)

k-Sets and Topological Invariants of Plane Curves

Suppose you and I each take a pen and draw a closed curve on a sheet of paper. Suppose we demand that the curves be regular (no cusps or corners), but they may be heavily self-intersecting.

We might look at the results and wonder whether we can somehow continuously transform one into the other. The answer will, of course, depend on what kind of transformations we allow. If all we care about is that the curve remains regular throughout the transformation, then an old theorem of Whitney's guarantees that the global winding number (the total number of turns that the tangent vector makes as we go around the curve) is a complete invariant: Two curves are can be transformed into one another iff they have the same winding number.

In the course of such a transformation, however, various degeneracies may occur, notably triple points or self-tangencies. If we forbid one or more of these degeneracies, then we get a much larger variety of ``non-equivalent'' curves. In the early 1990's, Arnol'd, inspired by the Vassiliev invariants from knot theory, introduced three numerical invariants as a first step toward classifying, or rather: distinguishing curves under these various notions of equivalence.

I would like to give a very brief and informal introduction to these invariants, and to analyze them for a particular family of curves that arise in the context of one of my favorite problems, the k-set problem. So far, this does not yield any improved asymptotic bounds, but I would like to make the case that this viewpoint is interesting and deserves further study.

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