The goals of this project were to study the relationship between, on the one hand, the geometry of non-Euclidean or fractal objects, and, on the other hand, the calculus of non-smooth curves and functions defined in these geometries. This was done from a few perspectives.

One strand, building on work of Cheeger and others, was to study non-smooth geometries that nonetheless admit a form of differential calculus generalizing the standard multivariable calculus taught to students. This provides insight into the foundations of calculus and permits the use of calculus tools in areas where the traditional techniques may not apply. In one paper with Kleiner and two with Kinneberg, the PI showed how these types of geometries are highly constrained in certain circumstances, and gave applications to problems in geometric measure theory and hyperbolic geometry.

Another topic in the proposal concerns curves of finite length in non-smooth geometries: what are nearly optimal paths to travel in these settings? In our standard geometry, theorems of Jones, Okikiolu, and Schul characterize such curves as connected sets that are “flat at most locations and scales”. With Schul, the PI gave a partial generalization of this to non-smooth geometries, although there are obstructions in this setting that are not seen in Euclidean geometry.

A large part of the project concerned finding “pieces” of smooth or Euclidean structure hidden within potentially non-smooth geometries or mappings, allowing us to bring powerful analytic techniques to bear on problems that might originally not seem amenable to them. This was the topic of work done during the project by the PI in a paper with Le Donne and two papers with Schul.

Lastly, a question one can ask of any non-Euclidean geometry is: can one view it, without too much distortion, as a subset of a standard Euclidean geometry, or another simpler geometry? The PI studied this from a theoretical perspective, but this question is now also of importance in computer science and data science. This was the subject of the PI’s work in two papers with Eriksson-Bique (where it was shown that this question may in fact have a negative answer in some not-too-complicated settings) and in papers with Vellis and Eriksson-Bique and Vellis (where it was shown that in the case of certain tree-like spaces that arise in geometry, one can always find such simplifications).

Over its four year lifespan, this project fully or partly funded work of the PI and collaborators on seven published papers and six additional preprint manuscripts that have been or will be submitted for publication. These include articles appearing in strong journals (e.g., Geometry & Topology and Advances in Mathematics), and articles solving open problems of Heinonen—Semmes (1997), Semmes (2001), and Azzam—Schul (2012). These articles were all posted publicly, and the PI gave talks on project-supported research at (live and online) venues around the world, including a plenary talk at the 4^th Northeastern Analysis Meeting at Syracuse University in 2019.

From a Broader Impacts perspective, the PI during the course of the project advised undergraduate and masters research projects on related contemporary topics. He co-organized a conference and an American Mathematical Society special session, and he also used the project to bring mathematics colloquium speakers to visit his home department.

Last Modified: 07/30/2021
Modified by: Guy C David