Sascha Troscheit

A dimension theory approach to embeddings in random geometry.

The continuum random tree and Brownian map are important metric spaces in probability theory and represent the "typical" tree and metric on the sphere, respectively. The Brownian map in particular is linked to Louville Quantum Gravity but the exact nature of the correspondence is unknown.
In this talk I will explain a fairly dynamical construction of these spaces and show how recent advances in the dimension theory of self-similar sets can be used to shed light on general embedding problems. In particular, I will show that the Assouad dimension of these metric spaces is infinite and how this restricts the nature of embeddings.
Time permitting, I will also indicate how the construction of continuum trees may be used to analyse highly singular functions such as the Weierstrass-type functions.