My area of interest is geometric measure theory. More precisely, I focus on quantitative aspects of rectifiability. Also, geometric problems from harmonic analysis, for example boundedness of singular integral operators on non-smooth sets.

Preprints

●   Ricci curvature lower bounds and uniform rectifiability

Joint work with M. Hyde and I.Y. Violo. arXiv preprint (2023).

●   Carleson's $\varepsilon^2$ conjecture in higher dimensions.

Joint work with I. Fleschler and X. Tolsa. arXiv preprint (2023).

●   Quantitative differentiability on uniformly rectifiable sets and applications to Sobolev trace theorems.

With J. Azzam and M. Mourgoglou. arXiv preprint (2023).

●   Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's $\varepsilon^2$ conjecture in higher dimensions

Joint with I. Fleschler and X. Tolsa . arXiv preprint (2023).

●   Analytic capacity and dimension of sets with plenty of big projections.

Joint with D. Dabrowski. arXiv preprint (2022).

●   Cone and paraboloid points of arbitrary subsets of Euclidean space

Joint with M. Hyde. arXiv preprint (2021). Warning! There is an error in the proof of the $C^{1,\alpha}$ result. The cone point result is correct. We hope to fix this soon.

●   Higher dimensional Jordan curves.

arXiv preprint (2020). Note: this was part of my PhD thesis. However I struggled to publish it because it has been misunderstood (poor boy). One criticism was good though: that the notion of "higher dimensional curve" is not sufficiently developed. I'm hoping to just do that as soon as I will find the time.

Published articles

●   Structure of sets with nearly maximal Favard length.

Joint with A. Chang, D. Dabrowski and T. Orponen. arXiv preprint (2022). Accepted for publication in Anal. & PDEs

●   Integrability of orthogonal projections, and applications to Furstenberg sets.

Joint with D. Dabrowski and T. Orponen. arXiv preprint (2021). Published in Adv. Math.

●   Sub-elliptic boundary value problems in flag domains.

Joint with T. Orponen. arXiv preprint (2020). Published in Adv. Calc. Var.

●   Necessary condition for the $L^2$ boundedness of the Riesz transform in Heisenberg group.

Joint with D. Dabrowski. arXiv preprint (2020). Published in Mat. Proc. Cam. Phil. Soc. (2023).

●   A proof of Carleson $\epsilon^2$ conjecture.

Joint with B. Jaye and X. Tolsa. arXiv preprint (2019). Published in Ann. of Math. (2021).

●   A square function involving the center of mass and rectifiability.

arXiv preprint (2019). Published in Math. Z. (2022).

●   Quantitative comparisons of multiscale geometric properties.

Joint with J. Azzam. Published in Anal. PDEs. (2021).

●   $\Omega$-symmetric measures and related singular integrals.

arXiv preprint (2019). Published in Rev. Mat. Iberoam. (2021).

●   Tangent points of lower content regular sets and $\beta$ numbers.

arXiv preprint (2017). Published in J. Lond. Math. Soc.

Notes

●   A note on Hausdorff dimension of sets with plenty of big projections.

arXiv preprint (2021). This is article is now overseded by "Analytic capacity and dimension of sets with plenty of big projection", with D. Dabrowski.