University of Minnesota

Title: Analysis and PDEs in domains with lower-dimensional boundaries

We shall discuss the recent progress at the interface of harmonic analysis, partial differential equations, and geometric measure theory, culminating in the full PDE characterization of regularity (uniform rectifiability) of sets, including the case of domains with lower dimensional boundaries.

University of Padua

Title: Lipschitz graphs

The theory of intrinsic Lipschitz graphs in Carnot groups was introduced about 15 years ago by B. Franchi, R. Serapioni and F. Serra Cassano. In this minicourse we plan to provide a brief account on some recent results, such as: extension and approximation results in general Carnot groups, Rademacher theorem (almost everywhere differentiability) for intrinsic Lipschitz graphs in Heisenberg groups, and some connections with the theory of Heisenberg currents.

NYU Courant

Title: Differentiability and rectifiability in the Heisenberg group

Differentiability and rectifiability describe how well functions and sets can be linearly approximated at infinitesimal scales. For many problems, knowing the behavior of a set at small scales isn't enough; one needs to quantify how well functions and sets can be approximated at many different scales. In this minicourse, we will explore quantitative differentiability and rectifiability in the nonlinear setting of the Heisenberg group and how they can be used to study surfaces in $\mathbb{H}$ and metric embeddings of $\mathbb{H}$.