**support the Oxfam petition**

**"Tax the Rich"**. This is a petition that asks for (great) wealth to be taxed in order to fund the ecological and social transition. On the topic, see the Oxfam report on

**climate equality**. The numbers are maddening (and this is a maths website!).

# Research

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).

● ** 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

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

Joint with D. Dabrowski. arXiv preprint (2022). To appear in TAMS.

● ** 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.

## Talks

Feb. 2022. Workshop talk at the Workshop on Harmonic analysis, Singular Integrals and PDEs, within the trimester proram "Interactions between Geometric measure theory, Singular integrals, and PDE" at HIM, Bonn.

Dec. 2021. Seminar talk at the Harmonic Analysis seminar of Universite Paris-Saclay.

Dec. 2021. Seminar talk at the Bilbao Analysis and PDEs seminar.

October 2021. Talk at Rajchman, Zygmund, Marcinkiewicz conference. IM PAN, Warsaw (Poland).

August 2021. Speaker at Geometric measure theory and applications in Cortona (Italy).

May 2021. Seminar talk at the Analysis group seminar at Oulu university.

Oct. 2020. Seminar talk at the Geometry Seminar of the University of Jyv\"{a}skyl\"{a}.

June 2020. Seminar talk at the Virtual Harmonic Analysis seminar. Online seminar, organised by a network of UK-based mathematicians.

Nov. 2019. Seminar talk at the Analysis seminar of the Universtiy of Edinburgh.

Oct. 2019. * A proof of the Carleson $\epsilon^2$-conjecture*. Talk at the workshop on Geometry and Analysis at IM PAN (Warsaw) in October 2019.

June 2019.
* $\Omega$-symmetric measures and related singular integrals*. Contributed talk at BAC2019 , June 2019.

June 2019. Contributed talk at the HAPDE conference in Helsinki (June 2019).

May 2019.
* A family of analyst's travelling salesman theorems *. Seminar talk at the joint Analysis Seminar of UAB-UB in Barcelona, 20/05/2019.

May 2017. * Towards a new characterisation of uniform rectifiability *. Talk given at the MIGSAA 2017 symposium.

April 2017. * Non-tangential behaviour and Carleson measures *. Talk given for the SMSTC (Scottish Mathematical Training Center) course of Harmonic Analysis.

June 2016. * Classification of three dimensional steady flow *. Talk given at the Topological fluid dynamics seminar at the University of Dundee following VI Arnold's work on the subject.

## Conferences, Workshops, Summer schools

- IAS/PCMI Graduate School in Harmonic Analysis. July 2018, Park City, Utah (US).
- Harmonic Analysis and Geometric measure theory. A conference at CIRM, Marseilles, France. October 2017.
- Neurogeometry. This was a summer school organised by SMI (Scuola matematica interuniversitaria) in Cortona, at the Palazzone della Normale. July 2017.
- New trends in Analysis and Geometry in Matric Spaces. A series of short courses organised by CIME-CIRM in Levico Terme, Italy. June 2017.
- Geometric PDEs at Warwick. A week-long workshop at the Mathematics Institute at Warwick University, UK. December 2017.
- SMI Summer School. A summer school for just-graduated students. Perugia, Italy. August 2016.

# Teaching

Under construction!

# Activism

Under construction!