Project IV 2020-2021

Square peg problems.

Andrew Lobb

Description

A Jordan curve is the image of an injective continuous map from the circle into the Euclidean plane. In 1912, Toeplitz asked the question whether there exist four points on the Jordan curve that form the vertices of a square (in other words whether every Jordan curve admits an inscribed square). This is known as the Square Peg Problem and it is still open.

Nevertheless there has been a variety of work on the problem over the past century or so. This project involves understanding and presenting some of this work.

Emch (1913) gave a solution for smooth convex Jordan curves, and this was built on by Schnirelmann (1929) who solved it for general smooth Jordan curves. In 1977 Meyerson gave an argument that showed there exists an inscribed rectangle in every continuous Jordan curve. Hugelmeyer improved on Meyerson's argument in 2018 to show (for example) that every smooth Jordan curve admits a rectangle with ratio of long edge to short edge being the square root of 3. Recently Greene and Lobb showed that every similarity class of rectangle is inscribed in every smooth Jordan curve.

One can also consider regularity conditions on the Jordan curve between continuous and smooth (see recent work of Tao and Feller-Golla), and also consider whether other quadrilaterals (apart from rectangles) might always be inscribed in every Jordan curve.

There are extensions to higher dimensions (for example inscribed solid shapes inside embeddings of the 2-sphere in 3-dimensional Euclidean space) to be considered. There are also related problems such as Fenn's table theorem (can one place a square table levelly on a floor that is not level everywhere) and Kronheimer-Kronheimer's work on the tripos problem.

A feature of much of the work on these problems seems to be that the arguments are cute and neat rather than long and technical.

There will be some common aspects to the project that we will explore as a group for most of the first term. After that there will be the possibility for specialization as you each pick aspects of the Square Peg Problem and its relatives that interest you.

Prerequisites

Topology III.

Co-requisites

Algebraic Topology IV.

Resources

You should take a look at the following:

  • There's "A Survey on the Square Peg Problem" by Matschke which appeared in the Notices of the AMS. You should be able to find a pdf by googling.
  • There's a YouTube video by 3Blue1Brown on the Rectangular Peg Problem.

email: Andrew Lobb.

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