# 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.

## Papers and preprints

Here is a list of papers, some published, some in preprint version.

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

Joint work with Damian Dabrowski.

We study sets with plenty of big projections (PBP). Among other things, we show that Vitushkin's conjecture on the relation between analytic capacity and Favard length holds for this class of sets and that wiggly sets with PBP have large dimension. Both results are obtained via Analyst's Travelling Salesman-type estimates: one (behind the former result) for an appropriately constructed Frostman measure, and the the other directly for the set (behind the latter).

Preprint (2022).

12. ** Structure of sets with nearly maximal Favard length. **

Joint work with Alan Chang, Damian Dabrowski and Tuomas Orponen.

We show that if a sets with finite Hausdorff one measure has very large Favard length (meaning that its Favard length to Hausdorff length ratio is very close to that of a segment), then it must be contained in a Lipschitz graph of small constant, save a portion of small measure.

Preprint (2022).

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

I show that an Analyst's traveling salesman theorem holds for sets with plenty of big projections. This gives some corollaries on the Hausdorff dimension of wiggly sets with PBP and lower bounds on their analytic/Lipschitz harmonic capacities. This is article is now overseded by "Analytic capacity and dimension of sets with plenty of big projection", with Damian Dabrowski (see #13).

Preprint (2021).

10. ** Integrability of orthogonal projections, and applications to Furstenberg sets. **

Joint work with Damian Dabrowski and Tuomas Orponen.

Given a measure with polynomial growth, we study the integrability properties of its push forwards along orthogonal projections. This gives some corollary about dimension of Furstenberg sets.

* Submitted.* Preprint (2021).

9. ** Cone and paraboloid points of arbitrary subsets of Euclidean space **

Joint work with Matthew Hyde.

We characterise tangent points of general subsets of Euclidean space without any assumption on densities. The methods also give $C^{1,\alpha}$-rectifiability results.

* Submitted. * Preprint (2021).

8. ** Sub-elliptic boundary value problems in flag domains. **

Joint work with Tuomas Orponen.

We study the solvability of the Dirichlet and Neumann problems for the sub-ellptic Kohn-Laplacian, where the boundary values are in $L^2$. We solve these problems in flag domains in $\mathbb{R}^3$ developing the method of layer potentials in Heisenberg geometry.

* Adv. Calc. Var. *To appear. Preprint (2020)

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

Joint with D. Dabrowski.

In this joint paper with Damian we prove an Heisenberg groups version of a result of Guy David. We show that, given a measure $\mu$, if the Heisenberg Riesz transform is bounded with respect to $\mu$, then $\mu$ has to have polynomial growth of the correct order.

* Submitted. * Preprint (2020).

6.** A proof of Carleson $\epsilon^2$ conjecture.**

Joint with Ben Jaye and Xavi Tolsa.

We prove an old conjecture of Carleson, originating from his work with C.J. Bishop, J. Garnett and P. Jones on harmonic measure. The result leads to a characterisation of
tangent points of Jordan curves in the plane via a Dini condition on the so-called Carleson $\varepsilon^2$ function.

* Ann. of Math. *
Vol. 194, No. 1 (July 2021), pp. 97-161 (65 pages). Article

5. ** A square function involving the center of mass and rectifiability. **

For a measure $\mu$ with finite upper denity a.e., I show that a Dini condition on the center of mass is necessary for the rectifiability of $\mu$. Together with a sufficiency result of Mayboroda and Nazarov, this gives a characterisation of rectifiability in terms of center of mass similar to that of Azzam and Tolsa for the $\beta_\mu$ coefficients.

* Math. Zeitschrift. *To appear. Preprint 2019.

4. ** Higher dimensional Jordan curves. **

I show that sets which have a certain topological stability satisfy a Jones' traveling salesman theorem type estimate. I obtain some corollaries, for example on the Hausdorff dimension of wiggly sets which also satisfy the topological condition.

* Submitted. * Preprint 2019.

3. ** Quantitative comparisons of multiscale geometric properties. **

Joint work with Jonas Azzam.

The starting question on this paper was the following. In the nineties, David and Semmes introduced
many multiscale geometric properties to quantifies the local regularity of sets and measures. Examples are: how close to a plane is the set
at location $x$ and scale $r$? How symmetric around a point? How convex? When the set under analysis is Ahlfors regular, David and Semmes showed that
all these properties have meaning and they are all 'synonyms'. Jonas and I asked: do these properties maintain a meaning beyond the Ahlfors regular regime?
An if so, are they still 'synonyms'?

* Anal. PDEs.* Vol. 14 (2021), No. 6, 1873–1904. Article.

2. ** $\Omega$-symmetric measures and related singular integrals. **

Let $S$ be the 1-sphere in the plane, and let \(\Omega: S \to S\) be bi-Lipschitz with constant $1+\delta_\Omega$, where $\delta_\Omega$
should be thought to be small (it will bounded above by a universal constant smaller than 1). In this note we prove that if an Ahlfors-David 1-regular measure $\mu$ is symmetric with respect to $ \Omega$, that is, if
$
\int_{B(x,r)} |x-y|\Omega\left(\frac{x-y}{|x-y|}\right) \, d\mu(y) = 0$ for all $x \in {\rm spt}(\mu)$ and $r>0$,
then $\mu$ is flat, or, in other words, there exists a constant $c>0$ and a line $L$ so that $\mu= c \mathcal{H}^{1}|_{L}$.
This result will be applied in a future companion paper to give a characterisation of rectifiability in terms of finitness of a certain square function involving this type of kernels.

* Rev. Mat. Iberoam. * Volume 37, Issue 5, 2021, pp. 1669–1715. Article.

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

Given a lower content $d$-regular set in $R^n$, I prove that the subset of points in $E$ where a certain Dini-type condition on the so-called Jones
$\beta$ numbers holds coincides with the set of tangent points of $E$, up to a set of $H^d$-measure zero.
The main point of our result is that $H^d|_E$ is not $\sigma$-finite; because of this, I use a certain variant of the $\beta$ coefficient, firstly introduced by Azzam and Schul, which is given in terms of integration with respect to the Hausdorff content.

* J. Lond. Math. Soc. * Volume101, Issue2, April 2020, Pages 530-555. Article.

## 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!

# Extra

Under construction!