Counting inscribed geometric objects
par
125
Braconnier
Enumerative geometric problems deal with counting geometric objects in a special position relative to another object and naturally appear in a number of fields. Simple examples in low-dimensional geometry include, for example, a
celebrated Toeplitz' square peg problem, or Pannwitz' and Denne's results on knot quadrisecants. In algebraic geometry one counts algebraic curves of a fixed degree and genus satisfying various passage/tangency conditions.
As usual in real geometry, in order for such numbers to be locally constant, one has to count the corresponding objects with certain signs. Corresponding signs are not always easy to find (in some cases they remain mysterious)
and even given these signs, proving that the total signed count is invariant under small deformations can be tedious.
I will discuss a number of various counting problems, including triangles inscribed in planar curves, bitangents and binormals of planar curves, characteristic bitangents of immersed hypersurfaces, and various secants of knots and links. All of them allow for a similar treatment based on maps of configuration spaces and the intersection theory.