The Rectangular Peg Problem and microlocal sheaf theory

The Square Peg Problem asks whether every Jordan curve in the plane contains four distinct points that form the vertices of a square. This
problem was proposed by Toeplitz in 1911 and remains unsolved in full generality. It can be generalized to the Rectangular Peg Problem, which concerns the existence of inscribed rectangles with a prescribed aspect ratio. Recently, Greene and Lobb successfully applied techniques in symplectic geometry to the problem and obtained new results. In this talk, I will explain how microlocal sheaf theory allows us to further extend their approach and affirmatively solve the Rectangular Peg Problem for a large class of Jordan curves, including all curves of finite length. This is joint work with Tomohiro Asano.