Differentiability in Carnot Groups and Metric Measure Spaces

A function is considered to be smooth or differentiable if at every point it is has a derivative, or in other words, a well-defined rate of change. Many familiar functions are smooth, and smoothness properties are convenient and prevalent in scientific applications. However, non-smooth functions also frequently arise in mathematics and its applications, such as optimization. This project concerns differentiability phenomena in non-smooth environments. Specifically, it seeks to understand when non-smooth objects possess hidden smoothness structures. While non-smooth objects are more difficult to understand, they are often equipped with additional structure that is not initially visible. For instance, Lipschitz functions (i.e., those functions which expand distances by at most a multiplicative factor) are differentiable at most points of their domain. The project investigates these and related phenomena, it seeks to describe when a partially defined function can be extended to a smooth function, and explores when a function can be approximated by a smooth function. The project will promote research collaboration and will generate research training opportunities for both graduate and undergraduate students. The project centers on two broad topics of research. First, the PI seeks a deeper understanding of the Whitney extension and Lusin approximation questions for mappings between Carnot groups. A significant complication, not present in the Euclidean case, is that the maps to be constructed must satisfy nonlinear constraints reflecting the underlying geometry of these non-Euclidean environments. A second line of study investigates the differentiability properties of Lipschitz functions in Euclidean spaces, Carnot groups, and metric or Banach spaces. A fundamental theorem due to Rademacher states that every Lipschitz function defined in a Euclidean domain is differentiable almost everywhere. However, in many situations one in fact finds differentiability points inside measure zero sets. This observation led to the modern study of sets of universal differentiability. The project seeks to test the limits of Rademacher’s theorem through an improved understanding of universal differentiability sets, via the use of maximal directional derivatives and other methods. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.