A classical question due to Toeplitz is the "square peg problem": given a simple closed curve in R^2, does there exist a set of four points on the curve forming a square? We present some new results on this (and similar) problems. An example theorem is this:

Let C be a closed C^1 curve in a Riemannian manifold, and x_1:x_2:....:x_n be a ratio so that each x_i is less than the sum of the remaining x_j. Then there exists a geodesic polygon inscribed in C with edgelengths in the ratio x_1:...:x_n.

This talk is joint work with Elizabeth Denne (Smith College) and John McCleary (Vassar College).

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