# Research

## Research summary

Research summary

### 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 measure theory, harmonic analysis, and PDEs.

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 measure theory, harmonic analysis, and PDEs.

## Publication list

Publication list

- [arXiv]
*Singular integrals on $C^{1,\alpha}$ regular curves in Carnot groups.*(with V. Chousionis and S. Li). Submitted. - [arXiv]
*Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces*. (with V. Chousionis, S. Li, and V. Vellis) Ann. Acad. Sci. Fenn. Math. To appear. - [arXiv]
*An implicit function theorem for Lipschitz mappings into metric spaces.*(with P. Hajlasz) Indiana Univ. Math. J. To appear. - [arXiv] [journal]
*A $C^m$ Whitney Extension Theorem for horizontal curves in the Heisenberg group*. (with A. Pinamonti and G. Speight) Trans Am Math Soc. 371 (2019), 8971-8992. - [arXiv] [journal]
*The Traveling Salesman Theorem in Carnot groups*. (with V. Chousionis and S. Li) Calc. Var. Partial Differential Equations. 58:14 (2019). - [arXiv] [journal]
*Weak BLD mappings and Hausdorff measure*. (with P. Hajlasz and S. Malekzadeh) Nonlinear Anal. 177 (2018), 524-531. - [arXiv] [journal]
*Sobolev extensions of Lipschitz mappings into metric spaces*. Int. Math. Res. Notes IMRN. 2019 (2019), 2241–2265. - [arXiv] [journal]
*The Whitney Extension Theorem for $C^1$, horizontal curves in the Heisenberg group.*J. Geo. Anal. 28 (2018), 61-83. - [website]
*The Heisenberg groups.*(with P. Hajlasz) Embeddings and Extrapolation X, Lecture notes Paseky spring schools in analysis (2017), MatfyzPress, 93-155. - [arXiv] [journal]
*The Dubovitskii-Sard theorem in Sobolev spaces.*(with P. Hajlasz) Indiana Univ. Math. J. 66 (2017), 705-723. - [arXiv] [journal]
*Geodesics in the Heisenberg group*. (with P. Hajlasz) Anal. Geom. Metr. Spaces 3 (2015), 325-337.

## Active and related research questions

Active and related research questions

### Whitney extensions into Carnot groups

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

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?

### BLD maps

BLD maps

- 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?

### Other questions

Other questions

- 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

## PhD Thesis

PhD Thesis

*Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem*.

Doctoral Dissertation, University of Pittsburgh (2017). [link]