Related papers: Combinatorial differential geometry and ideal Bian…

We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some…

Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and…

An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…

We generalize the Poincare-Hopf theorem sum_v i(v) = X(G) to vector fields on a finite simple graph (V,E) with Whitney complex G. To do so, we define a directed simplicial complex as a finite abstract simplicial complex equipped with a…

In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by…

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of…

We extend previous results on healthy derivative self-interactions for a Proca field to the case of a set of massive vector fields. We obtain non-gauge invariant derivative self-interactions for the vector fields that maintain the…

Gradient vector fields are fundamental objects from both theoretical and practical perspectives, since various phenomena can be modeled within this framework. The ``moduli space'' of such vector fields provides the foundation for describing…

In this short review we introduce group field theory, a particular class of random tensor models, which represents nowadays one of the candidates for a fundamental theory of quantum gravity. We insist on the combinatorial richness of…

A finite dimensional operator that commutes with some symmetry group admits quotient operators, which are determined by the choice of associated representation. Taking the quotient isolates the part of the spectrum supporting the chosen…

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds…

We prove an identity on a graph analogous to Bochner's identity on a Riemannian manifold. An auxiliary graph called the complete tangent graph intervenes in the term corresponding to Ricci curvature.

Motivated by the wild behavior of isolated essential singularities in complex analysis, we study singular complex analytic vector fields $X$ on arbitrary Riemann surfaces $M$. By vector field singularities we understand zeros, poles,…

Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of…

We describe the Betti numbers of the edge ideals $I(G)$ of uniform hypergraphs $G$ such that $I(G)$ has linear graded free resolution. We give an algebraic equation system and some inequalities for the components of the $f$--vector of the…

Let G be a finite connected simple graph. We define the moduli space of conformal structures on G. We propose a definition of conformally covariant operators on graphs, motivated by [25]. We provide examples of conformally covariant…

We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results…

We generalize double bracket vector fields, originally defined on semisimple Lie algebras, to Poisson manifolds equipped with a pseudo-Riemannian metric by utilizing a symmetric contravariant 2-tensor field. We extend the normal metric on…

We classify the algebraic combinatorial geometries of arbitrary field extensions of transcendence degree greater than 4 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and…

We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs…