differential geometry, geometric models of the visual cortex, topos theory
8
Scopus Publications
30
Scholar Citations
3
Scholar h-index
Scopus Publications
Critical concave-convex problems in Carnot groups Mattia Galeotti, Eugenio Vecchi Nonlinear Differential Equations and Applications, 2026 We consider a model Dirichlet problem with concave-convex and critical nonlinearity settled in Carnot groups. Our aim is to prove the existence of two positive solutions in the spirit of a famous result by Ambrosetti, Brezis and Cerami. To this aim we use a variational Perron method combined with proper estimates of a family of functions which are minimizers of the relevant Sobolev inequality. Due to the lack of boundary regularity, we also have to be careful while proving that the first solution found is a local minimizer in the proper topology.
The Cortical V1 Transform as a Heterogeneous Poisson Problem Alessandro Sarti, Mattia Galeotti, Giovanna Citti SIAM Journal on Imaging Sciences, 2024 Receptive profiles of V1 cortical cells are very heterogeneous and act by differentiating the stimulus image as operators changing from point to point. A lightness and color constancy image can be reconstructed as the solution of the associated inverse problem, that is a Poisson equation with heterogeneous differential operators. At the neural level the weights of short range connectivity constitute the fundamental solution of the Poisson problem adapted point by point. A first demonstration of convergence of the result towards homogeneous reconstructions is proposed by means of homogenisation techniques.
Cortically Based Optimal Transport Mattia Galeotti, Giovanna Citti, Alessandro Sarti Journal of Mathematical Imaging and Vision, 2022 We introduce a model for image morphing in the primary visual cortex V1 to perform completion of missing images in time. We model the output of simple cells through a family of Gabor filters and the propagation of the neural signal accordingly to the functional geometry induced by horizontal connectivity. Then we model the deformation between two images as a path relying two different outputs. This path is obtained by optimal transport considering the Wasserstein distance geodesics associated to some probability measures naturally induced by the outputs on V1. The frame of Gabor filters allows to project back the output path, therefore obtaining an associated image stimulus deformation. We perform a numerical implementation of our cortical model, assessing its ability in reconstructing rigid motions of simple shapes.
MODULI OF G-COVERS OF CURVES: GEOMETRY AND SINGULARITIES Mattia Galeotti Annales De L Institut Fourier, 2022 We analyze the singular locus and the locus of non-canonical singularities of the moduli space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>ℝ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> </mml:math> of curves with a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> -cover for any finite group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> . We show that non-canonical singularities are of two types: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> -curves, that is singularities lifted from the moduli space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover accent="true"> <mml:mi>𝕄</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>g</mml:mi> </mml:msub> </mml:math> of stable curves, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> </mml:math> -curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:math> , the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> </mml:math> -locus is empty, which is the first fundamental step in evaluating the Kodaira dimension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover accent="true"> <mml:mi>ℝ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> </mml:msub> </mml:math> .
Moduli spaces of abstract and embedded Kummer varieties Mattia Galeotti, Sara Perna International Journal of Mathematics, 2021 In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack [Formula: see text] of abstract Kummer varieties and the second one is the stack [Formula: see text] of embedded Kummer varieties. We will prove that [Formula: see text] is a Deligne-Mumford stack and its coarse moduli space is isomorphic to [Formula: see text], the coarse moduli space of principally polarized abelian varieties of dimension [Formula: see text]. On the other hand, we give a modular family [Formula: see text] of embedded Kummer varieties embedded in [Formula: see text], meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space [Formula: see text] of embedded Kummer surfaces and prove that it is obtained from [Formula: see text] by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: [Formula: see text] could be obtained from [Formula: see text] via a contraction for all [Formula: see text].
Singularities of moduli of curves with a Universal Root Mattia Galeotti Documenta Mathematica, 2017 In a series of recent papers, Chiodo, Farkas and Ludwig carry out a deep analysis of the singular locus of the moduli space of stable (twisted) curves with an \\ell -torsion line bundle. They show that for \\ell\\leq 6 and \\ell\\ne 5 pluricanonical forms extend over any desingularization. This opens the way to a computation of the Kodaira dimension without desingularizing, as done by Farkas and Ludwig for \\ell=2 , and by Chiodo, Eisenbud, Farkas and Schreyer for \\ell=3 . Here we treat roots of line bundles on the universal curve systematically: we consider the moduli space of curves C with a line bundle L such that L^{\\otimes\\ell}\\cong \\omega_C^{\\otimes k} . New loci of canonical and non-canonical singularities appear for any k\\not\\in\\ell\\Bbb Z and \\ell>2 , we provide a set of combinatorial tools allowing us to completely describe the singular locus in terms of dual graphs. We characterize the locus of non-canonical singularities, and for small values of \\ell we give an explicit description.
RECENT SCHOLAR PUBLICATIONS
Critical concave-convex problems in Carnot groups M Galeotti, E Vecchi arXiv preprint arXiv:2512.04640 , 2025 2025
A Heterogeneous Model of Boundary and Figure Completion in M Galeotti, G Citti, A Sarti International Conference on Geometric Science of Information, 205-214 , 2025 2025
Benamou-Brenier and Kantorovich on sub-Riemannian manifolds with no abnormal geodesics G Citti, M Galeotti, A Pinamonti arXiv preprint arXiv:2507.20959 , 2025 2025 Citations: 1
Critical singular problems in Carnot groups S Biagi, M Galeotti, E Vecchi arXiv preprint arXiv:2506.07521 , 2025 2025 Citations: 1
The cortical V1 transform as a heterogeneous poisson problem A Sarti, M Galeotti, G Citti SIAM Journal on Imaging Sciences 17 (1), 389-414 , 2024 2024 Citations: 3
An extension to non-nilpotent groups of Rothschild-Stein lifting method M Galeotti arXiv preprint arXiv:2403.19619 , 2024 2024 Citations: 1
Existence of a global fundamental solution for Hörmander operators M Galeotti arXiv e-prints, arXiv: 2403.19619 , 2024 2024
Differential Operators Heterogenous in Orientation and Scale in the Cortex M Galeotti, G Citti, A Sarti International Conference on Geometric Science of Information, 465-473 , 2023 2023
Cortically based optimal transport M Galeotti, G Citti, A Sarti Journal of Mathematical Imaging and Vision 64 (9), 1040-1057 , 2022 2022 Citations: 2
A framework for stereo vision via optimal transport M Galeotti, A Sarti, G Citti arXiv preprint arXiv:2207.00333 , 2022 2022
Moduli of -covers of curves: geometry and singularities M Galeotti Annales de l'Institut Fourier 72 (6), 2191-2240 , 2022 2022 Citations: 8
Birational geometry of moduli of curves with an S3-cover M Galeotti Advances in Mathematics 389, 107898 , 2021 2021 Citations: 4
Moduli spaces of abstract and embedded Kummer varieties M Galeotti, S Perna International Journal of Mathematics 32 (08), 2150054 , 2021 2021 Citations: 1
Kernel approximations in lie groups and application to group-invariant CNN G Lupi 2021 Citations: 1
Limiti e fallimenti dei modelli epidemiologici e previsionali nell’epidemia di SARS-COV-2 E Campo, G Gobo, M Galeotti, P Parra Saiani BENESSERE TECNOLOGIA SOCIETÀ, 39-48 , 2021 2021 Citations: 3
Límites y fallas de los modelos epidemiológicos y predictivos en la epidemia de SARS-COV-2 en Italia P Parra Saiani, E Campo, G Gobo, M Galeotti Impactos Sociales del Covid-19. Miradas desde la Sociología, 87-89 , 2021 2021 Citations: 1
Discipline: Mathématiques M Galeotti Université de Montpellier , 2017 2017
Moduli of curves with principal and spin bundles: singularities and global geometry MF Galeotti Université Pierre et Marie Curie-Paris VI , 2017 2017
Singularities of moduli of curves with a universal root M Galeotti Documenta Mathematica 22, 1337-1373 , 2017 2017 Citations: 4
MOST CITED SCHOLAR PUBLICATIONS
Moduli of -covers of curves: geometry and singularities M Galeotti Annales de l'Institut Fourier 72 (6), 2191-2240 , 2022 2022 Citations: 8
Birational geometry of moduli of curves with an S3-cover M Galeotti Advances in Mathematics 389, 107898 , 2021 2021 Citations: 4
Singularities of moduli of curves with a universal root M Galeotti Documenta Mathematica 22, 1337-1373 , 2017 2017 Citations: 4
The cortical V1 transform as a heterogeneous poisson problem A Sarti, M Galeotti, G Citti SIAM Journal on Imaging Sciences 17 (1), 389-414 , 2024 2024 Citations: 3
Limiti e fallimenti dei modelli epidemiologici e previsionali nell’epidemia di SARS-COV-2 E Campo, G Gobo, M Galeotti, P Parra Saiani BENESSERE TECNOLOGIA SOCIETÀ, 39-48 , 2021 2021 Citations: 3
Cortically based optimal transport M Galeotti, G Citti, A Sarti Journal of Mathematical Imaging and Vision 64 (9), 1040-1057 , 2022 2022 Citations: 2
Benamou-Brenier and Kantorovich on sub-Riemannian manifolds with no abnormal geodesics G Citti, M Galeotti, A Pinamonti arXiv preprint arXiv:2507.20959 , 2025 2025 Citations: 1
Critical singular problems in Carnot groups S Biagi, M Galeotti, E Vecchi arXiv preprint arXiv:2506.07521 , 2025 2025 Citations: 1
An extension to non-nilpotent groups of Rothschild-Stein lifting method M Galeotti arXiv preprint arXiv:2403.19619 , 2024 2024 Citations: 1
Moduli spaces of abstract and embedded Kummer varieties M Galeotti, S Perna International Journal of Mathematics 32 (08), 2150054 , 2021 2021 Citations: 1
Kernel approximations in lie groups and application to group-invariant CNN G Lupi 2021 Citations: 1
Límites y fallas de los modelos epidemiológicos y predictivos en la epidemia de SARS-COV-2 en Italia P Parra Saiani, E Campo, G Gobo, M Galeotti Impactos Sociales del Covid-19. Miradas desde la Sociología, 87-89 , 2021 2021 Citations: 1
Critical concave-convex problems in Carnot groups M Galeotti, E Vecchi arXiv preprint arXiv:2512.04640 , 2025 2025
A Heterogeneous Model of Boundary and Figure Completion in M Galeotti, G Citti, A Sarti International Conference on Geometric Science of Information, 205-214 , 2025 2025
Existence of a global fundamental solution for Hörmander operators M Galeotti arXiv e-prints, arXiv: 2403.19619 , 2024 2024
Differential Operators Heterogenous in Orientation and Scale in the Cortex M Galeotti, G Citti, A Sarti International Conference on Geometric Science of Information, 465-473 , 2023 2023
A framework for stereo vision via optimal transport M Galeotti, A Sarti, G Citti arXiv preprint arXiv:2207.00333 , 2022 2022
Discipline: Mathématiques M Galeotti Université de Montpellier , 2017 2017
Moduli of curves with principal and spin bundles: singularities and global geometry MF Galeotti Université Pierre et Marie Curie-Paris VI , 2017 2017
