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.

Publication list

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

  2. [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.

  3. [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.

  4. [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.

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

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

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

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

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

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

  11. [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