turkmath.org

In talk is about an open problem in plane geometry due to Otto Toeplitz (1911) and several related problems about inscribing and circumscribing polytopes in and around manifolds. Toeplitz originally asked whether every Jordan curve contains the four vertices of a square. Over the years it was proved in various special cases, nonetheless it remains an open problem in its full generality. We will also mention recent progress on the classification of quadrilaterals that can be inscribed in convex curves.