Traveling along Hölder curves

One goal of geometric measure theory is to understand a measure through its interaction with canonical lower dimension sets. The interaction of Radon measures in the plane or a higher-dimensional Euclidean space with finite sets or rectifiable curves is now completely understood. However, with respect to any other elementary family of sets, we only know how measures behave under additional regularity hypotheses. To make progress towards understanding the structure of Radon measures, we need to first understand the geometry of more classes of sets.

 

I will describe my latest work with L. Naples and V. Vellis, in which we find sufficient conditions to identify (subsets of) Hölder continuous curves of Hausdorff dimension $s>1$. Our conditions are related to the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves. On the other hand, standard self-similar sets such as the Sierpinski carpet show that our sufficient condition is not necessary. I will discuss this and other obstructions to the problem of characterizing Hölder curves and their subsets. The core of this talk requires very little background and will be accessible to graduate students, advanced undergraduates, as well as faculty working outside of analysis and geometry.