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Publications (26)

math.AP2024

A simple proof of the $1$ -dimensional flat chain conjecture

Andrea Marchese, Andrea Merlo

We give a new, elementary proof of the fact that metric 1-currents in the Euclidean space correspond to Federer-Fleming flat chains.

physics.plasm-ph2023

Physics-regularized neural network of the ideal-MHD solution operator in Wendelstein 7-X configurations

Andrea Merlo, Daniel BÃ¶ckenhoff, Jonathan Schilling +3

The computational cost of constructing 3D magnetohydrodynamic (MHD) equilibria is one of the limiting factors in stellarator research and design. Although data-driven approaches ha…

math.MG2024

Carnot rectifiability and Alberti representations

Gioacchino Antonelli, Enrico Le Donne, Andrea Merlo

A metric measure space is said to be Carnot-rectifiable if it can be covered up to a null set by countably many biLipschitz images of compact sets of a fixed Carnot group. In this…

math.MG2022

On the density problem in the parabolic space

Andrea Merlo, Mihalis Mourgoglou, Carmelo Puliatti

In this work we extend many classical results concerning the relationship between densities, tangents and rectifiability to the parabolic spaces, namely $R^{n + 1}$ equipped…

physics.plasm-ph2021

Proof of concept of a fast surrogate model of the VMEC code via neural networks in Wendelstein 7-X scenarios

Andrea Merlo, Daniel BÃ¶ckenhoff, Jonathan Schilling +7

In magnetic confinement fusion research, the achievement of high plasma pressure is key to reaching the goal of net energy production. The magnetohydrodynamic (MHD) model is used t…

math.MG2020

Marstrand-Mattila rectifiability criterion for $1$ -codimensional measures in Carnot Groups

Andrea Merlo

This paper is devoted to show that the flatness of tangents of $1$ -codimensional measures in Carnot Groups implies $C_{G}^{1}$ -rectifiability. As applications we prove that me…

math.CA2021

Endpoint Fourier restriction and unrectifiability

Giacomo Del Nin, Andrea Merlo

We show that if a measure of dimension $s$ on $R^{d}$ admits $(p, q)$ Fourier restriction for some endpoint exponents allowed by its dimension, namely $q = \frac{s}{d} p^{'}$ for…

math.CA2025

Marstrand's density theorem for arbitrary norms in the plane

Giacomo Del Nin, Andrea Merlo

We show that if a non-trivial measure in the plane admits, at almost every point, positive and finite $I^\pm$ -dimensional density with respect to some norm, then $I^\pm$ must be an integ…

math.MG2026

On the WALA conjecture, Alberti representations and applications

Andrea Merlo

The paper proves the WALA conjecture, showing that any AD‑regular Radon measure whose Lipschitz functions satisfy a weak affine approximation condition is uniformly rectifiable, an…

#uniform rectifiability#ad-regular measures#lipschitz functions#alberti representations

math.DG2025

Frobenius theorem and fine structure of tangency sets to non-involutive distributions

Giovanni Alberti, Annalisa Massaccesi, Andrea Merlo

In this paper we provide a complete answer to the question whether Frobenius' Theorem can be generalized to surfaces below the $C^{1, 1}$ threshold. We study the fine structure of t…

cs.LG2026

Improving ideal MHD equilibrium accuracy with physics-informed neural networks

Timo Thun, Andrea Merlo, Rory Conlin +2

We present a novel approach to compute three-dimensional Magnetohydrodynamic equilibria by parametrizing Fourier modes with artificial neural networks and compare it to equilibria…

math.MG2022

On rectifiable measures in Carnot groups: existence of density

Gioacchino Antonelli, Andrea Merlo

In this paper we start a detailed study of a new notion of rectifiability in Carnot groups: we say that a Radon measure is $P_{h}$ -rectifiable, for $h \in N$ , if it h…

cs.LG2025

ConStellaration: A dataset of QI-like stellarator plasma boundaries and optimization benchmarks

Santiago A. Cadena, Andrea Merlo, Emanuel Laude +8

Stellarators are magnetic confinement devices under active development to deliver steady-state carbon-free fusion energy. Their design involves a high-dimensional, constrained opti…

math.AP2025

Layer potentials for elliptic operators with DMO-type coefficients: big pieces $T b$ theorem, quantitative rectifiability, and free boundary problems

Andrea Merlo, Mihalis Mourgoglou, Carmelo Puliatti

For $n \geq 2$ , we consider the operator $L_{A} = - div (A (\cdot) \nabla)$ , where $A$ is a uniformly elliptic $(n + 1) \times (n + 1)$ matrix with variable coefficients, a Radon mea…

physics.plasm-ph2023

Accelerated Bayesian inference of plasma profiles with self-consistent MHD equilibria at W7-X via neural networks

Andrea Merlo, Andrea Pavone, Daniel BÃ¶ckenhoff +10

High- $⟨ I^^{2} ⟩$ operations require a fast and robust inference of plasma parameters with a self-consistent MHD equilibrium. Precalculated MHD equilibria are usually employ…

math.MG2022

On sets with unit Hausdorff density in homogeneous groups

Antoine Julia, Andrea Merlo

It is a longstanding conjecture that given a subset $E$ of a metric space, if $E$ has finite Hausdorff measure in dimension $I^\pm \geq 0$ and $H^{I^} \pm └ E$ has unit densi…

math.MG2020

Intrinsically Lipschitz functions with normal target in Carnot groups

Gioacchino Antonelli, Andrea Merlo

We provide a Rademacher theorem for intrinsically Lipschitz functions $I ¨ : U \subseteq W \to L$ , where $U$ is a Borel set, $W$ and $L$ are complementar…

math.AP2023

Generic uniqueness for the Plateau problem

Gianmarco Caldini, Andrea Marchese, Andrea Merlo +1

Given a complete Riemannian manifold $M \subset R^{d}$ which is a Lipschitz neighbourhood retract of dimension $m + n$ , of class $C^{h, I^^{2}}$ and an oriented, closed sub…

math.CA2021

Characterization of rectifiability via Lusin type approximation

Andrea Marchese, Andrea Merlo

We prove that a Radon measure $I^¼$ on $R^{n}$ can be written as $I^¼ = \sum_{i = 0}^{n} I^¼_{i}$ , where each of the $I^¼_{i}$ is an $i$ -dimensional rectifiable measure if and only if for…

math.MG2022

On rectifiable measures in Carnot groups: Marstrand-Mattila rectifiability criterion

Gioacchino Antonelli, Andrea Merlo

In this paper we continue the study of the notion of $P$ -rectifiability in Carnot groups. We say that a Radon measure is $P_{h}$ -rectifiable, for $h \in N$ ,…

math.MG2021

Geometry of $1$ -codimensional measures in Heisenberg groups

Andrea Merlo

This paper is devoted to the study of tangential properties of measures with density in the Heisenberg groups $H^{n}$ . Among other results we prove that measures with $(2n+1…

math.MG2021

On rectifiable measures in Carnot groups: representation

Gioacchino Antonelli, Andrea Merlo

This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $P$ -rectifiable measure. First, we show t…

math.FA2019

Full non-differentiability sets of typical Lipschitz functions

Andrea Merlo

In this paper we prove that the typical Lipschitz function has no directional derivative at any point of a Borel set $E$ if and only if $E$ is contained in a countable union of clo…

math.DG2025

Giovanni Alberti, Annalisa Massaccesi, Andrea Merlo

In this paper, we establish refined versions of the Frobenius Theorem for non-involutive distributions and use these refinements to prove an unrectifiability result for Carnot-Cara…

math.MG2021

Unextendable intrinsic Lipschitz curves

Gioacchino Antonelli, Andrea Merlo

In the setting of Carnot groups, we exhibit examples of intrinisc Lipschitz curves of positive $H^{1}$ -measure that intersect every connected intrinsic Lipschitz curve in a…

math.MG2022

On the converse of Pansu's Theorem

Guido De Philippis, Andrea Marchese, Andrea Merlo +2

We provide a suitable generalisation of Pansu's differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pa…

Publications (26)

A simple proof of the 1-dimensional flat chain conjecture

Physics-regularized neural network of the ideal-MHD solution operator in Wendelstein 7-X configurations

Carnot rectifiability and Alberti representations

On the density problem in the parabolic space

Proof of concept of a fast surrogate model of the VMEC code via neural networks in Wendelstein 7-X scenarios

Marstrand-Mattila rectifiability criterion for 1-codimensional measures in Carnot Groups

Endpoint Fourier restriction and unrectifiability

Marstrand's density theorem for arbitrary norms in the plane

On the WALA conjecture, Alberti representations and applications

Frobenius theorem and fine structure of tangency sets to non-involutive distributions

Improving ideal MHD equilibrium accuracy with physics-informed neural networks

On rectifiable measures in Carnot groups: existence of density

ConStellaration: A dataset of QI-like stellarator plasma boundaries and optimization benchmarks

Layer potentials for elliptic operators with DMO-type coefficients: big pieces Tb theorem, quantitative rectifiability, and free boundary problems

Accelerated Bayesian inference of plasma profiles with self-consistent MHD equilibria at W7-X via neural networks

On sets with unit Hausdorff density in homogeneous groups

Intrinsically Lipschitz functions with normal target in Carnot groups

Generic uniqueness for the Plateau problem

Characterization of rectifiability via Lusin type approximation

On rectifiable measures in Carnot groups: Marstrand-Mattila rectifiability criterion

Geometry of 1-codimensional measures in Heisenberg groups

On rectifiable measures in Carnot groups: representation

Full non-differentiability sets of typical Lipschitz functions

Tangency sets of non-involutive distributions and unrectifiability in Carnot-CarathÃ©odory spaces

Unextendable intrinsic Lipschitz curves

On the converse of Pansu's Theorem

A simple proof of the $1$ -dimensional flat chain conjecture

Marstrand-Mattila rectifiability criterion for $1$ -codimensional measures in Carnot Groups

Layer potentials for elliptic operators with DMO-type coefficients: big pieces $T b$ theorem, quantitative rectifiability, and free boundary problems

Geometry of $1$ -codimensional measures in Heisenberg groups