# Splitting Loops and Necklaces: Variants of the Square Peg Problem

## Florian Frick, CMU

## Date and time: 6pm, Tuesday, April 30, 2019

## Place: Courant Institute, WWH1314

In 1911 Toeplitz conjectured that any simple continuous closed curve
in the plane inscribes a square. A less famous variant of this problem
is Hadwiger's 1971 conjecture that any simple closed continuous curve
in 3-space inscribes a parallelogram. Both conjectures have been
resolved under some smoothness condition on the curve. I will survey
some of the known results and then report on recent progress on both
conjectures. In particular we resolve Hadwiger's conjecture in full
generality by relating it to partition results for real-valued
functions.

This is joint work with Jai Aslam, Shujian Chen, Sam Saloff-Coste,
Linus Setiabrata, and Hugh Thomas.