Pietro Fré
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Pietro Fré has a total of 116 co-authors. Here are the top co-authors by number of shared publications:
- Alexander S. Sorin
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia
28 shared publications - Mario Trigiante
Dipartimento DISAT Politecnico di Torino Torino Italy
17 shared publications - Pietro Antonio Grassi
Università del Piemonte Orientale
14 shared publications - Sergio Ferrara
Theoretical Physics Department CERN CH 1211 Geneva 23 Switzerland
12 shared publications Mario Trigiante
8 shared publicationsR. D'AUria
8 shared publications
Latest Publications
- Open Access
Thermodynamics à la Souriau on Kähler Non-Compact Symmetric Spaces for Cartan Neural Networks
In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a clear-cut distinction between the generalized thermodynamics associated with Integrable Dynamical Systems and the challenging proposal of Gibbs probability distributions on provided by generalized thermodynamics à la Souriau. Our main result is the proof that .s supporting such Gibbs distributions are only the Kähler ones. Furthermore, for the latter, we solve the problem of determining the space of temperatures, namely, of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of of a positivity domain in the Cartan subalgebra of the maximal compact subalgebra . We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi–Vesentini manifolds utilizing Paint Group symmetry. Furthermore, we claim that Rao’s, Chentsov’s, and Amari’s Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. In particular, we provide an explicit study of thermodynamical geometry for the Poincaré plane. The key feature of the Gibbs probability distributions in this setup is their covariance under the entire group of symmetries U. The partition function is invariant against transformations, and the set of its arguments, namely the generalized temperatures, can always be reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group .Open AccessThermodynamics à la Souriau on Kähler Non-Compact Symmetric Spaces for Cartan Neural Networks
In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a clear-cut distinction between the generalized thermodynamics associated with Integrable Dynamical Systems and the challenging proposal of Gibbs probability distributions on provided by generalized thermodynamics à la Souriau. Our main result is the proof that .s supporting such Gibbs distributions are only the Kähler ones. Furthermore, for the latter, we solve the problem of determining the space of temperatures, namely, of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of of a positivity domain in the Cartan subalgebra of the maximal compact subalgebra . We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi–Vesentini manifolds utilizing Paint Group symmetry. Furthermore, we claim that Rao’s, Chentsov’s, and Amari’s Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. In particular, we provide an explicit study of thermodynamical geometry for the Poincaré plane. The key feature of the Gibbs probability distributions in this setup is their covariance under the entire group of symmetries U. The partition function is invariant against transformations, and the set of its arguments, namely the generalized temperatures, can always be reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group . - 30 January 2026
- Published by Walter de Gruyter GmbH in Transport Phenomena
- Vol. 1 (1)
Axial symmetric Navier Stokes equations and the Beltrami/antiBeltrami spectrum in view of Physics Informed Neural Networks
In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply revised and developed in 2022 with M. Trigiante. Instead of a compact 3-torus T 3 = R 3 / Λ ${T}^{3}={\mathbb{R}}^{3}/{\Lambda}$ where Λ is a crystallographic lattice, as done in previous work, here I considered flows confined in a cylinder with identified opposite bases. In this topology I considered axial symmetric flows and found a complete basis of axial symmetric harmonic 1-forms that, for each energy level, decomposes into six components: two Beltrami, two antiBeltrami and two closed forms. These objects, that are written in terms of trigonometric and Bessel functions, constitute a function basis for an L 2 space of axial symmetric flows. I have presented a general scheme for the search of axial symmetric solutions of Navier Stokes equation by reducing the latter to a hierachy of quadratic relations on the development coefficients of the flow in the above described functional basis. It is proposed that the coefficients can be determined by means of a Physics Informed like Neural Network optimization recursive algorithm. Indeed the present paper provides the theoretical foundations for such an algorithmic construction that is planned for a future publication.- 30 January 2026
- Published by Walter de Gruyter GmbH in Transport Phenomena
- Vol. 1 (1)
Axial symmetric Navier Stokes equations and the Beltrami/antiBeltrami spectrum in view of Physics Informed Neural Networks
In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply revised and developed in 2022 with M. Trigiante. Instead of a compact 3-torus T 3 = R 3 / Λ ${T}^{3}={\mathbb{R}}^{3}/{\Lambda}$ where Λ is a crystallographic lattice, as done in previous work, here I considered flows confined in a cylinder with identified opposite bases. In this topology I considered axial symmetric flows and found a complete basis of axial symmetric harmonic 1-forms that, for each energy level, decomposes into six components: two Beltrami, two antiBeltrami and two closed forms. These objects, that are written in terms of trigonometric and Bessel functions, constitute a function basis for an L 2 space of axial symmetric flows. I have presented a general scheme for the search of axial symmetric solutions of Navier Stokes equation by reducing the latter to a hierachy of quadratic relations on the development coefficients of the flow in the above described functional basis. It is proposed that the coefficients can be determined by means of a Physics Informed like Neural Network optimization recursive algorithm. Indeed the present paper provides the theoretical foundations for such an algorithmic construction that is planned for a future publication.- Open Access
Tessellation Groups, Harmonic Analysis on Non-compact Symmetric Spaces and the Heat Kernel in view of Cartan Convolutional Neural Networks
In this paper, we continue the development of the Cartan neural networks programme, launched with three previous publications, by focusing on some mathematical foundational aspects that we deem necessary for our next steps forward. The mathematical and conceptual results are diverse and span various mathematical fields, but the inspiring motivation is unified. The aim is to introduce layers that are mathematically modeled as non-compact symmetric spaces, each mapped onto the next one by solvable group homomorphisms. In particular, in the spirit of Convolutional neural networks, we have introduced the notion of Tits Satake (TS) vector bundles where the TS submanifold is the base space. Within this framework, the tiling of the base manifold, the representation of bundle sections using harmonics, and the need for a general theory of separator walls motivated a series of mathematical investigations that produced both definite and partial results. Specifically, we present the group theoretical construction of the separators for all non-compact symmetric spaces $\mathrm{U/H}$, as well as of the $Δ_{8,3,2}$ tiling group and its normal Fuchsian subgroups, respectively yielding the uniformization of the genus $g=3$ Fermat Quartic and of the genus $g=2$ Bolza surface. The quotient automorphic groups are studied. Furthermore, we found a new representation of the Laplacian Green function and the Heat Kernel on Hyperbolic Spaces $\mathbb{H}^{n}$, and a setup for the construction of the harmonic functions in terms of the spinor representation of pseudo-orthogonal groups. Finally, to obtain an explicit construction of the Laplacian eigenfunctions on the Bolza Riemann surface, we propose and conjecture a new strategy relying on the Abel-Jacobi map of the Riemann surface to its Jacobian variety and the Siegel Theta function.Open AccessTessellation Groups, Harmonic Analysis on Non-compact Symmetric Spaces and the Heat Kernel in view of Cartan Convolutional Neural Networks
In this paper, we continue the development of the Cartan neural networks programme, launched with three previous publications, by focusing on some mathematical foundational aspects that we deem necessary for our next steps forward. The mathematical and conceptual results are diverse and span various mathematical fields, but the inspiring motivation is unified. The aim is to introduce layers that are mathematically modeled as non-compact symmetric spaces, each mapped onto the next one by solvable group homomorphisms. In particular, in the spirit of Convolutional neural networks, we have introduced the notion of Tits Satake (TS) vector bundles where the TS submanifold is the base space. Within this framework, the tiling of the base manifold, the representation of bundle sections using harmonics, and the need for a general theory of separator walls motivated a series of mathematical investigations that produced both definite and partial results. Specifically, we present the group theoretical construction of the separators for all non-compact symmetric spaces $\mathrm{U/H}$, as well as of the $Δ_{8,3,2}$ tiling group and its normal Fuchsian subgroups, respectively yielding the uniformization of the genus $g=3$ Fermat Quartic and of the genus $g=2$ Bolza surface. The quotient automorphic groups are studied. Furthermore, we found a new representation of the Laplacian Green function and the Heat Kernel on Hyperbolic Spaces $\mathbb{H}^{n}$, and a setup for the construction of the harmonic functions in terms of the spinor representation of pseudo-orthogonal groups. Finally, to obtain an explicit construction of the Laplacian eigenfunctions on the Bolza Riemann surface, we propose and conjecture a new strategy relying on the Abel-Jacobi map of the Riemann surface to its Jacobian variety and the Siegel Theta function. - Open Access
Navigation through Non-Compact Symmetric Spaces: a mathematical perspective on Cartan Neural Networks
Recent work has identified non-compact symmetric spaces U/H as a promising class of homogeneous manifolds to develop a geometrically consistent theory of neural networks. An initial implementation of these concepts has been presented in a twin paper under the moniker of Cartan Neural Networks, showing both the feasibility and the performance of these geometric concepts in a machine learning context. The current paper expands on the mathematical structures underpinning Cartan Neural Networks, detailing the geometric properties of the layers and how the maps between layers interact with such structures to make Cartan Neural Networks covariant and geometrically interpretable. Together, these twin papers constitute a first step towards a fully geometrically interpretable theory of neural networks exploiting group-theoretic structures
Open AccessNavigation through Non-Compact Symmetric Spaces: a mathematical perspective on Cartan Neural Networks
Recent work has identified non-compact symmetric spaces U/H as a promising class of homogeneous manifolds to develop a geometrically consistent theory of neural networks. An initial implementation of these concepts has been presented in a twin paper under the moniker of Cartan Neural Networks, showing both the feasibility and the performance of these geometric concepts in a machine learning context. The current paper expands on the mathematical structures underpinning Cartan Neural Networks, detailing the geometric properties of the layers and how the maps between layers interact with such structures to make Cartan Neural Networks covariant and geometrically interpretable. Together, these twin papers constitute a first step towards a fully geometrically interpretable theory of neural networks exploiting group-theoretic structures
- Open Access
Cartan Networks: Group theoretical Hyperbolic Deep Learning
Hyperbolic deep learning leverages the metric properties of hyperbolic spaces to develop efficient and informative embeddings of hierarchical data. Here, we focus on the solvable group structure of hyperbolic spaces, which follows naturally from their construction as symmetric spaces. This dual nature of Lie group and Riemannian manifold allows us to propose a new class of hyperbolic deep learning algorithms where group homomorphisms are interleaved with metric-preserving diffeomorphisms. The resulting algorithms, which we call Cartan networks, show promising results on various benchmark data sets and open the way to a novel class of hyperbolic deep learning architectures.
Open AccessCartan Networks: Group theoretical Hyperbolic Deep Learning
Hyperbolic deep learning leverages the metric properties of hyperbolic spaces to develop efficient and informative embeddings of hierarchical data. Here, we focus on the solvable group structure of hyperbolic spaces, which follows naturally from their construction as symmetric spaces. This dual nature of Lie group and Riemannian manifold allows us to propose a new class of hyperbolic deep learning algorithms where group homomorphisms are interleaved with metric-preserving diffeomorphisms. The resulting algorithms, which we call Cartan networks, show promising results on various benchmark data sets and open the way to a novel class of hyperbolic deep learning architectures.
- Open Access
Lectures on resolutions à la Kronheimer of orbifold singularities, McKay quivers for Gauge Theories on D3 branes, and the issue of Ricci flat metrics on the resolved three-folds
The present Lecture Notes have been prepared to back up a series of a few seminars given by the author at the Albert Einstein Institute in Potsdam. These Notes aim at reviewing a research project conducted over the last six years about a quite interesting and challenging topic, namely the use of the generalized Kronheimer construction and the generalized McKay correspondence for the crepant resolution $Y^Γ_{[3]} \to \frac{\mathbb{C}^3}Γ$ of orbifold singularities, $Γ$ being s finite subgroup of $\mathrm{SU(3)}$, as a strategic tool to construct holographic dual pairs of $\mathcal{N}=1$ gauge theories in $D=4$ and D3-brane solutions of type IIB supergravity. The project, developed by Ugo Bruzzo with the present author, in various co-authorships with Massimo Bianchi, Anna Fino, Pietro Antonio Grassi, Dimitry Markushevich, Dario Martelli, Mario Trigiante and Umar Shahzad, is still going on and there are a lot of interesting unanswered questions that are illustrated throughout these Lecture Notes and particularly emphasized in its last chapter. The mathematical roots of this project are very robust and ramified in differential geometry, group theory and algebraic geometry. A particularly challenging issue is that of constructing Ricci flat metrics on the total space of line-bundles over compact Kähler manifolds of real dimension four. In that direction the symplectic action/angle formalism turns out to be a winning weapon. The ultimate goal is that of deducing all the aspects of holographic duality from finite group theory or, to be more precise, SU(3) finite subgroupsOpen AccessLectures on resolutions à la Kronheimer of orbifold singularities, McKay quivers for Gauge Theories on D3 branes, and the issue of Ricci flat metrics on the resolved three-folds
The present Lecture Notes have been prepared to back up a series of a few seminars given by the author at the Albert Einstein Institute in Potsdam. These Notes aim at reviewing a research project conducted over the last six years about a quite interesting and challenging topic, namely the use of the generalized Kronheimer construction and the generalized McKay correspondence for the crepant resolution $Y^Γ_{[3]} \to \frac{\mathbb{C}^3}Γ$ of orbifold singularities, $Γ$ being s finite subgroup of $\mathrm{SU(3)}$, as a strategic tool to construct holographic dual pairs of $\mathcal{N}=1$ gauge theories in $D=4$ and D3-brane solutions of type IIB supergravity. The project, developed by Ugo Bruzzo with the present author, in various co-authorships with Massimo Bianchi, Anna Fino, Pietro Antonio Grassi, Dimitry Markushevich, Dario Martelli, Mario Trigiante and Umar Shahzad, is still going on and there are a lot of interesting unanswered questions that are illustrated throughout these Lecture Notes and particularly emphasized in its last chapter. The mathematical roots of this project are very robust and ramified in differential geometry, group theory and algebraic geometry. A particularly challenging issue is that of constructing Ricci flat metrics on the total space of line-bundles over compact Kähler manifolds of real dimension four. In that direction the symplectic action/angle formalism turns out to be a winning weapon. The ultimate goal is that of deducing all the aspects of holographic duality from finite group theory or, to be more precise, SU(3) finite subgroups - 24 July 2023
- Published by Walter de Gruyter GmbH
- p. 416-443
16 Isometries and the geometry of coset manifolds
16 Isometries and the geometry of coset manifolds was published in Discrete, Finite and Lie Groups on page 416.
- 24 July 2023
- Published by Walter de Gruyter GmbH
- p. 416-443
16 Isometries and the geometry of coset manifolds
16 Isometries and the geometry of coset manifolds was published in Discrete, Finite and Lie Groups on page 416.
- 24 July 2023
- Published by Walter de Gruyter GmbH
- p. 479-496
A Available MATHEMATICA NoteBooks written by the author
A Available MATHEMATICA NoteBooks written by the author was published in Discrete, Finite and Lie Groups on page 479.
- 24 July 2023
- Published by Walter de Gruyter GmbH
- p. 479-496
A Available MATHEMATICA NoteBooks written by the author
A Available MATHEMATICA NoteBooks written by the author was published in Discrete, Finite and Lie Groups on page 479.
- 24 July 2023
- Published by Walter de Gruyter GmbH
- p. 225-252
9 Monodromy groups of differential equations
9 Monodromy groups of differential equations was published in Discrete, Finite and Lie Groups on page 225.
- 24 July 2023
- Published by Walter de Gruyter GmbH
- p. 225-252
9 Monodromy groups of differential equations
9 Monodromy groups of differential equations was published in Discrete, Finite and Lie Groups on page 225.
- Open Access
- 24 July 2023
- Published by Walter de Gruyter GmbH
Some remarks about notations
Some remarks about notations was published in Discrete, Finite and Lie Groups on page XIII.
- 24 July 2023
- Published by Walter de Gruyter GmbH
Open AccessSome remarks about notations
Some remarks about notations was published in Discrete, Finite and Lie Groups on page XIII.