Date: Sunday, 13 Iyar 5765 (May 22, '05)

Speaker: Uli Wagner (Hebrew University)

Title: "k-Sets and Topological Invariants of Plane Curves"

Abstract:
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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 regularly
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 1992, Arnold, 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.