Related papers: Vector Fields, Invariant Varieties and Linear Syst…
We investigate the coupling between the inflaton and massive vector fields. All renormalizable couplings with shift symmetry of the inflaton are considered. The massive vector can be decomposed into a scalar mode and a divergence-free…
We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for…
We consider the general nonvanishing, divergence-free vector fields defined on a domain in three space and tangent to its boundary. Based on the theory of finite type invariants, we define a family of invariants for such fields, in the…
We consider a family of $C^1$ vector fields satisfying a suitable higher order involutivity condition. We discuss the definition of commutators, the regularity of Sussmann's orbits and the Poincar\'e inequality.
We focus on working on incidence rings, a class of (possibly infinite) matrix rings indexed by ordered sets. Some general properties about them are given, including how they are always the inverse limit of finite matrix rings, giving a…
This paper is based on my talk at ICM on recent progress in a number of classical problems of linear algebra and representation theory, based on new approach, originated from geometry of stable bundles and geometric invariant theory.
We give a new and self-contained proof of the existence and unicity of the flow for an arbitrary (not necessarily homogeneous) smooth vector field on a real supermanifold, and extend these results to the case of holomorphic vector fields on…
We study holomorphic vector fields whose singular locus contains a local complete intersection smooth positive-dimensional component. We prove global and local formulas expressing the limiting Milnor/Poincare-Hopf contribution along such a…
A vector topology on a vector space over a topological field is a (not necessarily Hausdorff) topology by which the addition and scalar multiplication are continuous. We prove that, if an isomorphism between the lattice of topologies of two…
A new cohomology, induced by a vector field, is defined on pairs of differential forms ($1$--differentiable forms) in a manifold. It is proved a link with the classical de Rham cohomology and an $1$-differentable cohomology of Lichnerowicz…
We consider a finite permutation group acting naturally on a vector space $V$ over a field $\Bbbk$. A well known theorem of G\"obel asserts that the corresponding ring of invariants $\Bbbk[V]^G$ is generated by invariants of degree at most…
For a large class of systems of o.d.e.'s which have first integrals, the method of arrays yields the following results: i) The first integrals $I$ can be found by solving systems of linear equations. ii) How the first integral $I$ responds…
The bivariate difference field provides an algebraic framework for a sequence satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order, and consider the multivariate difference…
We consider complex rational vector fields in dimension $n>2$ (equivalently, differential forms of degree $n-1$ in $n$ variables) which admit a Liouvillian first integral. Extending a classical result by Singer for $n=2$, our main result…
We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen…
We study the stable pairs theory of local curves in 3-folds with descendent insertions. The rationality of the partition function of descendent invariants is established for the full local curve geometry (equivariant with respect to the…
We outline an algorithm for construction of functional bases of absolute invariants under the rotation group for sets of rank 2 tensors and vectors in the Euclidean space of arbitrary dimension. We will use our earlier results for symmetric…
We give a global geometric decomposition of continuously differentiable vector fields on $\mathbb{R}^n$. More precisely, given a vector field of class $\mathcal{C}^{1}$ on $\mathbb{R}^{n}$, and a geometric structure on $\mathbb{R}^n$, we…
We combine Deligne's global invariant cycle theorem, and the algebraicity theorem of Cattani, Deligne and Kaplan, for the connected components of the locus of Hodge classes, to conclude that under simple assumptions these components are…
A large class of real $3$-dimensional nilpotent polynomial vector fields of arbitrary degree is considered. The aim of this work is to present general properties of the discrete and continuous dynamical systems induced by these vector…