## Publication list

1. [arXiv] Identifying 1-rectifiable measures in Carnot groups (with M. Badger and S. Li). Submitted.

2. [arXiv] Whitney's Extension Theorem and the finiteness principle for curves in the Heisenberg group. Submitted.

3. [arXiv] Singular integrals on $C^{1,\alpha}$ regular curves in Carnot groups (with V. Chousionis and S. Li). J. Anal. Math. To appear.

4. [arXiv] [journal] Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces (with V. Chousionis, S. Li, and V. Vellis) Ann. Acad. Sci. Fenn. Math. 45 (2020), 931–955.

5. [arXiv] [journal] An implicit function theorem for Lipschitz mappings into metric spaces (with P. Hajlasz) Indiana Univ. Math. J. 69 No. 1 (2020), 205-228.

6. [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.

7. [arXiv] [journal] The Traveling Salesman Theorem in Carnot groups (with V. Chousionis and S. Li) Calc. Var. Partial Differential Equations. 58:14 (2019).

8. [arXiv] [journal] Weak BLD mappings and Hausdorff measure (with P. Hajlasz and S. Malekzadeh) Nonlinear Anal. 177 (2018), 524-531.

9. [arXiv] [journal] Sobolev extensions of Lipschitz mappings into metric spaces. Int. Math. Res. Notes IMRN. 2019 (2019), 2241–2265.

10. [arXiv] [journal] The Whitney Extension Theorem for $C^1$, horizontal curves in the Heisenberg group. J. Geo. Anal. 28 (2018), 61-83.

11. [website] The Heisenberg groups (with P. Hajlasz) Embeddings and Extrapolation X, Lecture notes Paseky spring schools in analysis (2017), MatfyzPress, 93-155.

12. [arXiv] [journal] The Dubovitskii-Sard theorem in Sobolev spaces (with P. Hajlasz) Indiana Univ. Math. J. 66 (2017), 705-723.

13. [arXiv] [journal] Geodesics in the Heisenberg group (with P. Hajlasz) Anal. Geom. Metr. Spaces 3 (2015), 325-337.

## 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?

### 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

• 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?

## 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]

A geodesic path in the Heisenberg group