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Francesco Vaccarino

Publications and source records attributed to Francesco Vaccarino.

33 records · Page 2Linked to original sources
arXiv2016

The Nori-Hilbert scheme is not smooth for 2-Calabi Yau algebras

Let $k$ be an algebraically closed field of characteristic zero and let $A$ be a finitely generated $k-$algebra. The Nori - Hilbert scheme of $A$, parameterizes left ideals of codimension $n$ in $A,$ and it is well known to be smooth when $A$ is formally smooth. In this paper we will study the Nori - Hilbert scheme for $2-$Calabi Yau algebras. The main examples of these are surface group algebras and preprojective algebras. For the former we show that the Nori-Hilbert scheme is smooth for $n=1$ only, while for the latter we show that the smooth components that contain simple representations are precisely those that only contain simple representation. Under certain conditions we can generalize this last statement to arbitrary $2-$Calabi Yau algebras.

math.AG↗
arXiv2016

Persistent Homology analysis of Phase Transitions

Persistent homology analysis, a recently developed computational method in algebraic topology, is applied to the study of the phase transitions undergone by the so-called XY-mean field model and by the phi^4 lattice model, respectively. For both models the relationship between phase transitions and the topological properties of certain submanifolds of configuration space are exactly known. It turns out that these a-priori known facts are clearly retrieved by persistent homology analysis of dynamically sampled submanifolds of configuration space.

cond-mat.stat-mech↗
arXiv2015

Combinatorial resolutions of multigraded modules and multipersistent homology

Let $ R=k[x_1...x_r]$ and $M$ a multigraded $R-$module. In this work we interpret $M$ as a multipersistent homology module and give a multigraded resolution of it. The construction involves cellular resolutions of monomial ideals and reflects the combinatorial structure of multipersistence homology modules. In the one critical case, a multifiltration is represented by a labelled cellular complex. A multipersistence homology module measures the defect of acyclicity of the associated multigraded cellular chain complex.

math.AT↗
arXiv2015

A presentation of general multipersistence modules computable in polynomial time?

Multipersistence homology modules were introduced by G.Carlsson and A.Zomorodian which gave, together with G.Singh, an algorithm to compute their Groebner bases. Although their algorithm has polynomial complexity when the chain modules are free, i.e. in the one-critical case, it might be exponential in general. We give a new presentation of multipersistence homology modules, which allows us to design an algorithm to compute their Groebner bases always in polynomial time by avoiding the mapping telescope.

math.AT↗
arXiv2015

A new family of algebras whose representation schemes are smooth

We give a necessary and sufficient smoothness condition for the scheme parameterizing the n-dimensional representations of a finitely generated associative algebra over an algebraically closed field of characteristic zero. In particular, our result implies that the points M of the above scheme, satisfying $Ext_A ^2(M,M)=0$, are regular. This generalizes well-known results on finite-dimensional algebras to finitely generated algebras.

math.AG↗
arXiv2015

$P$-persistent homology of finite topological spaces

Let $P$ be a finite poset. We will show that for any reasonable $P$-persistent object $X$ in the category of finite topological spaces, there is a $P-$ weighted graph, whose clique complex has the same $P$-persistent homology as $X$.

math.AT↗
arXiv2014

Combinatorial presentation of multidimensional persistent homology

A multifiltration is a functor indexed by $\mathbb{N}^r$ that maps any morphism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in particular that the $\mathbb{N}^r$-graded $R[x_1,\ldots, x_r]$-modules that can occur as $R$-spans of multifiltrations of sets are the direct sums of monomial ideals.

math.AT↗
arXiv2013

Topological strata of weighted complex networks

The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and --more recently-- correlations between neighboring nodes. However, statistical methods quickly become cumbersome when dealing with many-body properties and do not capture the precise mesoscopic structure of complex networks. Here we introduce a novel method, based on persistent homology, to detect particular non-local structures, akin to weighted holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchically nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasilocal network properties, because of the intrinsic non-locality of homological properties, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. Moreover, this new method creates the first bridge between network theory and algebraic topology, which will allow to import the toolset of algebraic methods to complex systems.

cond-mat.stat-mech↗
arXiv2009

Moduli of linear representations, symmetric products and the non commutative Hilbert scheme

In this survey paper we study the relationships between the coarse moduli space which parameterizes the finite dimensional linear representations of an associative alegebra, the non commutative hilbert scheme and the affine scheme which is the spectrum of the abelianization of algebra of the divided powers. In particular we will show a map which specialize to the Hlibert - Chow morphism when the associative algebra is commutative. The extension to the positive characteristic case of some of the results due to L. Le Bruyn on noncommutative desingularization is outlined. The possibility to use our construction to extend the work done by C.H.Liu and S.T.Yau on D-Branes to their more recent work on non commutative case is underlined.

math.AG↗
arXiv2008

Hilbert-Chow morphism for non commutative Hilbert schemes and moduli spaces of linear representations

Let $k$ be a commutative ring and let $R$ be a commutative $k-$algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a (non necessarily commutative) $R-$algebra $A. $ We focus on the scheme $\ran//\GL_n$ of the $n-$dimensional representations of $A, $ on the Hilbert scheme $\Hilb_A^n$ parameterizing the left ideals of codimension $n$ of $A$ and on the affine scheme Spec $Γ_R^n(A)^{ab} $ of the abelianization of the divided powers of order $n$ over $A. $ We give a generalization of the Grothendieck-Deligne norm map from $\Hilb_A^n$ to Spec $Γ_R^n(A)^{ab} $ which specializes to the Hilbert Chow morphism on the geometric points when $A$ is commutative and $k$ is an algebraically closed field. Describing the Hilbert scheme as the base of a principal bundle we shall factor this map through the moduli space $\ran//\GL_n$ giving a nice description of this Hilbert-Chow morphism, and consequently proving that it is projective.

math.AG↗
arXiv2007

Symmetric products, linear representations and the commuting scheme

We show that the ring of multisymmetric functions over a commutative ring is isomorphic to the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices. As a consequence we give a surjection from the ring of invariants of several matrices to the ring of multisymmetric functions generalizing a classical result of H.Weyl and F.Junker. We also find a surjection from the ring of invariants over the commuting scheme to the ring of multisymmetric functions. This surjection is an isomophism over a characteristic zero field and induces an isomorphism at the level of reduced structures over an infinite field of positive characteristic.

math.AG↗
arXiv2004

Homogeneous Multiplicative Polynomial Laws are Determinants

Let R be a ring and let B be a commutative ring. Let p be a homogeneous multiplicative polynomial law of degree n from R to B. We show that p is essentially a determinant, in the sense that p is obtained from a determinant by left and right composition with ring homomorphisms.

math.RA↗
arXiv2004

On the Invariants of Matrices and the Embedding Problem

Let K be an infinite field and let R be a K-algebra endowed with a homogeneous polynomial norm N of degree n. If N satisfies a formal analogue of the Cayley-Hamilton Theorem the we will show that R is a quotient of the ring of the invariants of several square matrices of order n under the simultaneous conjugation.

math.RA↗
arXiv2004

The ring of multisymmetric functions

Let R be a commutative ring and let n,m be two positive integers. Let be the polynomial ring in m x n commuting independent variables R. The symmetric group on n letters acts diagonally on A(n,m). We give generators and relations of the rings of invariants for this action.

math.RA↗