Mathematics

I will discuss joint work with Guy C. David and Riku Anttila about a conjecture posed by Kleiner on the combinatorial Loewner property. With Anttila we recently discovered that the conjecture fails, and gave a new construction of interesting Loewner spaces and spaces which are not quasisymmetric to any Loewner space. With Guy C. David we discovered another curious example, called the pillow space. We show that if this pillow space is qusisymmetric to a Loewner space, then an analytically one dimensional sphere exists. This leads to the question in the title.  The goal of this talk is to advertise this problem, to show how it is connected to other conjectures and to showcase this recent work of our group.