My research focuses on analysis in metric spaces (often in Carnot groups such as the Heisenberg group) and its relationship with the geometry of the space. More generally, I study geometric function theory, geometric measure theory, and harmonic analysis in metric settings.
Active and related research questions
Whitney extensions into Carnot groups
When can we construct a smooth horizontal surface in the Heisenberg groups with prescribed derivative data on a compact subset?
Can we make such a construction in more general Carnot groups? Which ones? What additional data is required?
What if we only consider curves? How regular can these curves be?
When such a smooth horizontal surface or curve can be built, how much control do we have on the norms of the derivatives?
Quantitative rectifiability in Carnot groups
What is the relationship between the geometry of a set in a Carnot group and the boundedness of a singular integral operator restricted to that set?
In particular, what regularity is required for a curve in a Carnot group to guarantee the boundedness of a large class of singular integral operators on it?
Can we classify these regular curves in terms of the bounded singular integral operators?
To what degree can we quantify the approximation of regular sets in Carnot groups by affine sets?
Can we embed any doubling metric space into a Euclidean space with a map which controls the lengths of curves?
What kind of geometric and topological properties must such length-controlling mappings possess?
Can we build a non-trivial winding map in the Heisenberg group? If so, how bad can it be?
How does the branch set of such a mapping look?
Which results from classical analysis have a natural generalization to the setting of Lipschitz mappings from Euclidean space into metric spaces?
Which subsets of the Heisenberg group can be embedded into a Euclidean space in a bi-Lipschiz way?
When can a Lipschitz mapping between metric spaces be extended to a Sobolev map with a controlled norm?
A geodesic path in the Heisenberg group