arXiv research

Equivalence of second order differential operators in vector bundles studied.

Study non-commutative function algebras using contact geometry.

The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie pseudo-group of local feedback transformations. In particular, considered class d…

Study natural invariants for differential operators, simplifying their equivalence problem.

Study natural invariants for third order nonlinear operators on 2D manifolds.

We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U(1) coefficients and differential forms. A…

The paper classifies symbols of differential operators on vector bundles.

An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of …

The description of invariants of surfaces with respect to the motion groups is reduced to the description of invariants of parameterized surfaces with respect to the motion groups. Existence of a commuting system of invariant partial differential operators (derivatives) and a finite system of invariants, such that any …

Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.

Study of $q$-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.

New 3-manifolds bound rational 4-balls through specific operations.

The paper describes spectra of operators on rational homogeneous varieties.

We compare the star surgery operations introduced in [KS] to the generalized rational blow-down. We show that star surgery shares the properties that make rational blow-down useful for constructions of small exotic symplectic 4-manifolds. Then we show that star surgery operations provide a strictly more general class o…

Study eigenvalues of curvature operators to annihilate cobordism invariants.

The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.

Researchers found differential invariants for Kundt spacetimes.

Survey on minimal rational curves and their geometric structures.

Homology of abelian differentials stabilizes with more zeros.

Study of rational curves in complex manifolds with specific normal bundles.

The study shows conditions for Kähler manifolds to have rational cohomology of complex projective space.

Consider the space $R_Δ$ of rational functions of several variables with poles on a fixed arrangement $Δ$ of hyperplanes. We obtain a decomposition of $R_Δ$ as a module over the ring of differential operators with constant coefficients. We generalize to the space $R_Δ$ the notions of principal part and of residue, and …

Paper describes a new method for character varieties of surface groups.

Proves the Hodge conjecture for complex projective manifolds.

New method detects projective equivalences and symmetries in rational 3D curves.

Algorithm finds Liouvillian solutions for planar rational vector fields.

We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.

An ordinary differential field $(F,d)$ of characteristic zero, a subgroup $H$ of affine group $GL(n,C)\propto C^n$ with respect to its identical representation in $F^n$ and the following two fields of differential rational functions in $x=(x_1,x_2,...,x_n)$-column vector, $$C< x, d>^H=\{f^d< x> \in C< x, d> : f^d< hx+…

For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier-Douady, 3-class, w, on the manifold, we define the ring of differential operators `acting on sections of the projective bundle' in a formal sense. In particular, any oriented even-dimensional manifold carries a projective s…

We verify that the rational blow-down schemes along certain Seifert fibered 3-manifolds found by the second author, Szabo and Wahl are, in fact, symplectic operations.

We classify the rational differential 1-forms with simple poles and simple zeros on the Riemann sphere according to their isotropy group; when the 1-form has exactly two poles the isotropy group is isomorphic to $\mathbb{C}^{*}$, namely $\{z\mapsto az\ \vert\ a\in\mathbb{C}, a\neq0\}$, and when the 1-form has $k\geq 3$…

New metrics compare rational spectra using optimal transport.

The study proves conditions for convex hypersurfaces in Riemannian manifolds to be rational homology spheres.

The paper connects curvature positivity to rational connectedness in complex geometry.

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of…

Motivated by a result of L.P. Roberts on rational blow-downs in Heegaard-Floer homology, we study such operations along 3-manifolds that arise as branched double covers of $S^{3}$ along several non-alternating, slice knots.

We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.

The study shows conditions for Kähler manifolds to have rational cohomology.

We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the averaged system disappears, due to the cooperation of the generators. We investi…

Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.

An differential field $(F;\partial_1,...,\partial_m)$ of characteristic zero, a subgroup $H$ of affine group $GL(n,C)\propto C^n$ with respect to its identical representation in $F^n$ and the following two fields of differential rational functions in $x=(x_1,x_2,...,x_n)$-column vector, $$C< x, \partial >^H=\{f^{\part…

Let $H$ be a hyperexponential function in $n$ variables $x=(x_1,\dots,x_n)$ with coefficients in a field $\mathbb{K}$, $[\mathbb{K}:\mathbb{Q}] <\infty$, and $ω$ a rational differential $1$-form. Assume that $Hω$ is closed and $H$ transcendental. We prove using Schanuel conjecture that there exist a univariate function…

We compute the Hilbert polynomial and the Poincare function counting the number of fixed jet-order differential invariants of conformal metric structures modulo local diffeomorphisms, and we describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action. This r…

Let h be a rationally even cohomology theory and h^ the natural differential refinement, as defined by Hopkins and Singer. We consider the possible definitions of the relative differential cohomology groups, generalizing the analogous picture for the Deligne cohomology, and we show the corresponding long exact sequence…

Boring is an operation which converts a knot or two-component link in a 3--manifold into another knot or two-component link. It generalizes rational tangle replacement and can be described as a type of 2--handle attachment. Sutured manifold theory is used to study the existence of essential spheres and planar surfaces …

In this paper it is shown that multiplicative cohomology theories that are rationally even -- a technical condition that is often satisfied -- the Hopkins-Singer construction of generalized differential cohomology has a unital, graded commutative multiplicative structure. To this end, an explicit integration and a diff…

Suppose that $C=(C_1,..., C_m)$ is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold $(X,ω)$ with connected, negative definite intersection graph $Γ_C$. We show that by replacing an appropriate neighborhood of $\cup C_i$ with a smoothing $W_S$ of a normal surface singularity $(S, 0)$ w…

Paper introduces rational Gaussian wavelets for efficient signal approximation.