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Maurizio Parton

Publications and source records attributed to Maurizio Parton.

At least 37 records · Page 2Linked to original sources
arXiv2019

SAI, a Sensible Artificial Intelligence that plays Go

We propose a multiple-komi modification of the AlphaGo Zero/Leela Zero paradigm. The winrate as a function of the komi is modeled with a two-parameters sigmoid function, so that the neural network must predict just one more variable to assess the winrate for all komi values. A second novel feature is that training is based on self-play games that occasionally branch -- with changed komi -- when the position is uneven. With this setting, reinforcement learning is showed to work on 7x7 Go, obtaining very strong playing agents. As a useful byproduct, the sigmoid parameters given by the network allow to estimate the score difference on the board, and to evaluate how much the game is decided.

cs.AI↗
arXiv2018

The Role of Spin(9) in Octonionic Geometry

Starting from the 2001 Thomas Friedrich's work on Spin(9), we review some interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin(9) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley-Rosenfeld planes and to three series of Grassmannians.

math.DG↗
arXiv2018

Structure of locally conformally symplectic Lie algebras and solvmanifolds

We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional connected and simply connected solvable Lie groups.

math.DG↗
arXiv2018

Primitive weird numbers having more than three distinct prime factors

In this paper we study some structure properties of primitive weird numbers in terms of their factorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms for generating weird numbers having a given number of distinct prime factors are presented. These algorithms yield primitive weird numbers of the form $mp_1\dots p_k$ for a suitable deficient positive integer $m$ and primes $p_1,\dots,p_k$ and generalize a recent technique developed for generating primitive weird numbers of the form $2^np_1p_2$. The same techniques can be used to search for odd weird numbers, whose existence is still an open question.

math.NT↗
arXiv2018

Primitive abundant and weird numbers with many prime factors

We give an algorithm to enumerate all primitive abundant numbers (briefly, PANs) with a fixed $Ω$ (the number of prime factors counted with their multiplicity), and explicitly find all PANs up to $Ω=6$, count all PANs and square-free PANs up to $Ω=7$ and count all odd PANs and odd square-free PANs up to $Ω=8$. We find primitive weird numbers (briefly, PWNs) with up to 16 prime factors, improving the previous results of [Amato-Hasler-Melfi-Parton] where PWNs with up to 6 prime factors have been given. The largest PWN we find has 14712 digits: as far as we know, this is the largest example existing, the previous one being 5328 digits long [Melfi]. We find hundreds of PWNs with exactly one square odd prime factor: as far as we know, only five were known before. We find all PWNs with at least one odd prime factor with multiplicity greater than one and $Ω= 7$ and prove that there are none with $Ω< 7$. Regarding PWNs with a cubic (or higher) odd prime factor, we prove that there are none with $Ω\le 7$, and we did not find any with larger $Ω$. Finally, we find several PWNs with 2 square odd prime factors, and one with 3 square odd prime factors. These are the first such examples.

math.NT↗
arXiv2017

Almost complex structures on spheres

In this paper we review the well-known fact that the only spheres admitting an almost complex structure are S^2 and S^6. The proof described here uses characteristic classes and the Bott periodicity theorem in topological K-theory. This paper originates from the talk "Almost Complex Structures on Spheres" given by the second author at the MAM1 workshop "(Non)-existence of complex structures on S^6", held in Marburg from March 27th to March 30th, 2017. It is a review paper, and as such no result is intended to be original. We tried to produce a clear, motivated and as much as possible self-contained exposition.

math.DG↗
arXiv2016

Clifford systems in octonionic geometry

We give an inductive construction for irreducible Clifford systems on Euclidean vector spaces. We then discuss how this notion can be adapted to Riemannian manifolds, and outline some developments in octonionic geometry.

math.DG↗
arXiv2015

The even Clifford structure of the fourth Severi variety

The Hermitian symmetric space $M=\mathrm{EIII}$ appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure. This means the existence of a real oriented Euclidean vector bundle $E$ over it together with an algebra bundle morphism $φ:\mathrm{Cl}^0(E) \rightarrow \mathrm{End}(TM)$ mapping $Λ^2 E$ into skew-symmetric endomorphisms, and the existence of a metric connection on $E$ compatible with $φ$. We give an explicit description of such a vector bundle $E$ as a sub-bundle of $\mathrm{End}(TM)$. From this we construct a canonical differential 8-form on $\mathrm{EIII}$, associated with its holonomy $\mathrm{Spin}(10) \cdot \mathrm{U}(1) \subset \mathrm{U}(16)$, that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at $\mathrm{EIII}$ as the smooth projective variety $V_{(4)} \subset \mathbb{C}P^{26}$ known as the fourth Severi variety.

math.DG↗
arXiv2013

Spin(9) geometry of the octonionic Hopf fibration

We deal with Riemannian properties of the octonionic Hopf fibration S^{15}-->S^8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S^1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.

math.DG↗
arXiv2012

Holomorphic submersions of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold X admits a holomorphic submersion with positive dimensional fibers at least one of which is of Kähler type, then X is globally conformally Kähler or biholomorphic, up to finite covers, to a Vaisman manifold (i.e. a mapping torus over a circle, with Sasakian fibre). As a consequence, we show that the product between a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.

math.DG↗
arXiv2012

Spheres with more than 7 vector fields: all the fault of Spin(9)

We give an interpretation of the maximal number of linearly independent vector fields on spheres in terms of the Spin(9) representation on R^16. This casts an insight on the role of Spin(9) as a subgroup of SO(16) on the existence of vector fields on spheres, parallel to the one played by complex, quaternionic and octonionic structures on R^2, R^4 and R^8, respectively.

math.DG↗
arXiv2011

Spin(9) and almost complex structures on 16-dimensional manifolds

For a Spin(9)-structure on a Riemannian manifold M^16 we write explicitly the matrix psi of its Kähler 2-forms and the canonical 8-form Phi. We then prove that Phi coincides up to a constant with the fourth coefficient of the characteristic polynomial of psi. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Phi and Phi^2 in the special case of holonomy Spin(9).

math.DG↗
arXiv2010

Examples of non-trivial rank in locally conformal Kähler geometry

We consider locally conformal Kaehler geometry as an equivariant, homothetic Kaehler geometry (K,Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γto its dilation factors, thus completing the description of locally conformal Kaehler geometry in this equivariant setting. The rank r of a locally conformal Kaehler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that r is non-trivial, providing explicit examples of locally conformal Kaehler manifolds with 1<r<b_1. As far as we know, these are the first examples of this kind. Moreover, we prove that locally conformal Kaehler Oeljeklaus-Toma manifolds have either r=b_1 or r=b_1/2.

math.DG↗
arXiv2005

Locally conformal parallel $G_2$ and $Spin(7)$ manifolds

We characterize compact locally conformal parallel $G_2$ (respectively, $Spin(7)$) manifolds as fiber bundles over $S^1$ with compact nearly Kähler (respectively, compact nearly parallel $G_2$) fiber. A more specific characterization is provided when the local parallel structures are flat.

math.DG↗
arXiv2005

Reduction of Vaisman structures in complex and quaternionic geometry

We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kaehler manifold (K,Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kaehler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover we define locally conformal hyperkaehler reduction as an equivariant version of hyperkaehler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally we show that locally conformal hyperkaehler reduction induces hyperkaehler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperkaehler reduction.

math.DG↗
arXiv2002

Families of strong KT structures in six dimensions

This paper classifies Hermitian structures on 6-dimensional nilmanifolds M=G/L for which the fundamental 2-form is d d-bar closed, a condition that is shown to depend only on the underlying complex structure J of M. The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obtained from a limacon-shaped curve in the complex plane. Related theory provides examples of various types of Ricci-flat structures.

math.DG↗
arXiv2002

Locally conformal Kaehler reduction

We define reduction of locally conformal Kaehler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We show the compatibility between this reduction and Kaehler reduction of the universal cover. By a recent result of Kamishima and the second author, in the Vaisman case (that is, when a metric in the conformal class has parallel Lee form) if the manifold is compact its universal cover comes equipped with the structure of Kaehler cone over a Sasaki compact manifold. We show the compatibility between our reduction and Sasaki reduction, hence describing a subgroup of automorphisms whose action causes reduction to bear a Vaisman structure. Then we apply this theory to construct a wide class of Vaisman manifolds.

math.DG↗